General theory of conical flows and its application to supersonic aerodynamics

MISSING IMAGE

Material Information

Title:
General theory of conical flows and its application to supersonic aerodynamics
Series Title:
NACA TM
Physical Description:
333 p. : ill. ; 27 cm.
Language:
English
Creator:
Germain, Paul, 1920-
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Superposition principle (Physics)   ( lcsh )
Harmonic functions   ( lcsh )
Aerodynamics   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The report deals with a method of studying the equation of cylindrical waves particularly indicated for the solution of certain aerodynamic problems. The method reduces problems of a hyperbolic equation to problems of harmonic functions. The study has been applied toward setting up the fundamental principles, to developing their investigation up to calculation of the pressures on the visualized obstacles, and to showing how the initial field of "conical flows" was considerably enlarged by a procedure of integral superposition.
Bibliography:
Includes bibliographic references (p. 261-263).
Statement of Responsibility:
by Paul Germain.
General Note:
"Report date January 1955."
General Note:
"Translation of "La théorie générale des mouvements coniques et ses applications a l̕ aérodynamique supersonique." Office National ď Études et de Recherches Aéronautiques, no. 34, 1949."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778920
oclc - 44857672
sobekcm - AA00006145_00001
System ID:
AA00006145:00001

Full Text
nCA--M -'31







qq^ 74l2'- q1-7-167S




















GENERAL THEORY OF CONICAL FLOWS AND ITS APPLICATION

TO SUPERSONIC AERODYNAMICS




By

Paul Germain




Preface

By

M. J. Peres





NOTICE


This report deals with a method of studying the equation of cylin-
drical waves particularly indicated for the solution of certain problems
in aerodynamics. One of the most remarkable aspects of this method is
that it reduces problems of a hyperbolic equation to problems of harmonic
functions. We have applied ourselves here to setting up the fundamental
principles, to developing their investigation up to calculation of the
pressures on the visualized obstacles, and to showing how the initial
field of "conical flows" was considerably enlarged by a procedure of
integral superposition.

Such an undertaking entails certain dangers. In France the exist-
ence of conical flows was not known before 1946. Abroad, this question
has, for a long time, given rise to numerous reports which either were
not published or were published only after a certain delay. Thus it
must be pointed out that some of the results here obtained, original in
France when found, doubtlessly were not original abroad. Nevertheless
it seems possible to me to specify a certain number of points treated
in this report which, even considering the lapse of time, appear as new:
the parts concerning homogeneous flows, the general study of conical
flows with infinitesimal cone angles, the numerical or analogous methods
for the study of flows flattened in one direction, and a certain number
of the results of chapter IV. Moreover, even where the results which we
found independently were already known abroad, the employed methods are
not always identical.

Another peculiarity should be noted. Since these questions actually
are everywhere the object of numerous investigations, progress has made
very rapid strides. This report edited at the beginning of 1948, risks
appearing, in certain aspects, slightly outmoded in 1949. To extenuate
this inconvenience we have indicated in a brief appendix placed at the
end of this report the progress made in these questions during the last
year. This appendix is followed by a supplementary bibliography which
indicates recent reports concerning our subject, or older ones of which
we had no previous knowledge.

I should not have been able to successfully terminate this report
without the advice and support of my teacher, Mr. J. Peres, and it is
very important to me to express here my great respect for and gratitude
to him.

I should equally cite all those who directly or less directly have
contributed to my intellectual development and to whom I owe so much:
my teachers of special mathematics and of normal school, Mr. Bouligand
who directed my first reports, Mr. Villat, promoter of the Study of the
Mechanics of Fluids in France whose brilliant instruction has been of
the greatest value to me.








I also feel obliged to thank the directors of the O.N.E.R.A. who
have facilitated my task, and especially Mr. Girerd, director of aero-
dynamic research.






PREFACE


With his research on conical flows and their application, Mr. Paul
Germain has made a major contribution to the very timely study of super-
sonic aerodynamics. The present volume offers a comprehensive expose
which had been still lacking, an expose of elegance and solid construc-
tion containing a number of original developments. The author has fur-
thermore considered very thoroughly the applications and has shown how
one may solve within the scope of linear theory, by combinations of
conical flows, the general problems of the supersonic wing, taking into
account dihedral and sweepback, and also fuselage and control surface
effects. The analysis he develops in this respect leads him to methods
which permit, either by calculation alone or with the support of
electrolytic-tank experimentation, complete and accurate numerical
determinations.

After a few -preliminary developments (particularly on the validity
of the hypothesis of linearization), chapter I is devoted to the gener-
alities concerning conical flows. In such flows the velocity components
depend only on two variables and their determination makes use of har-
monic functions or of functions which verify the wave equation with two
variables according to whether one is inside or outside of the Mach
cone. Mr. Germain specifies the conditions of agreement between func-
tions defined in one domain or in the other and shows that the study of
conical flows amounts in general to boundary problems relative to three
analytical functions connected by differential relationships. He studies,
on the other hand, homogeneous flows which generalize the cone flows and
are no less useful in the applications.

From the viewpoint of the linear theory of supersonic flows one
must maintain two principal types of conical flows, bounded respectively
by an obstacle in the form of a cone with infinitesimal cone angle, and
by an obstacle in the form of a cone flattened in one direction.

The general investigation of the flows of the first type is entirely
Mr. Germain's own and forms the object of chapter II of his book. By a
subtle analysis of the approximations which may be legitimate Mr. Germain
succeeds in simplifying the rather complex boundary problem he had to
deal with; he replaces it by an external Hilbert problem. He shows how
it is possible, after having obtained the solution for an orientation
of the cone in the relative air stream, to pass, in a manner as simple
as it is elegant, to the calculation of the effect of a change in inci-
dence. He gives general formulas for the forces, treats completely
diverse noteworthy special cases and finally applies the method of trigo-
nometric operators which is also his own to the practical numerical
calculation of the flow about an arbitrary cone.

The determination of movements about infinitely flattened cones has
formed the object of numerous reports. The analysis which Mr. Germain
develops for this question (chapter III) contributes simplifications,







specifications, and important supplements. Thus he evolves, in the case
of an obstacle inside the Mach cone, a principle of minimum singularity
which enters into the determination of the solution. Mr. Germain gives
two original methods for treatment of the general case: one utilizes
the electrolytic-tank analogy, surmounting the difficulty arising from
the experimental application of the principle of minimum singularity;
the other, purely numerical, involves the trigonometric operators quoted
above.

In the last chapter, finally, Mr. Germain visualizes the composi-
tion of conical flows with regard to aerodynamic calculation of a super-
sonic aircraft. Concerning this subject he develops a complete theory
which covers most of the known results and incorporates new ones. He
concludes with an outline of the flows past a flat dihedral, with appli-
cation to the fins and control surfaces.

The creation of the National Office for Aeronautical Study and
Research has already made possible the setting up of groups of investi-
gators which do excellent work in several domains that are of interest
to modern aviation and put us on the level of the best research centers
abroad. Mr. Paul Germain inspirits and directs one of those groups in
the most efficient manner. He is one of those, and the present report
will suffice to bear out this statement, on whom we can count for the
development of the study of aerodynamics in France.



Joseph Peres
Member of the Academy of Sciences





NACA TM 1554


TABLE OF CONTENTS


Pages

CHAPTER I GENERALITIES ON CONICAL FLOWS . 1
1.1 Equations of Supersonic Linearized Flows ... 1
1.2 Generalities on Conical Flows . .. 10
1.5 Homogeneous Flows . . ... ... .22

CHAPTER II CONICAL FLOWS WITH INFINITESIMAL CONE ANGLES ..... 30
2.1 Solution of the Problem . ... 30
2.2 Applications . . ... 41
Cone of Revolution . . ... .44
Elliptic Cone ....... . .. ..... 47
Study of a Cone With Semicircular Section . .. .58
2.5 Numerical Calculation of Conical Flows With
Infinitesimal Cone Angles . . .. .62
Calculation of the Trigonometric Operators ... 68

CHAPTER III CONICAL FLOWS INFINITELY FLATTENED IN ONE
DIRECTION . . . ... 79
5.1 Cone Obstacle Entirely Inside the Mach Cone ... 80
Study of the Elementary Problems (Symmetrical Cone ..... 80
Flows With Respect to Oxlx)) . .. ..80
Nonsymmetrical Conical Flows . .97
General Problem . . ... ...... 105
Rheo-Electric Method . .... .108
Purely Numerical Method ................. 117
5.2 Case Where the Cone Is Not Inside the Mach Cone () 152
Cone Totally Bisecting the Mach Cone (Fig. 28) ... 154
Cone Partially Inside and Partially Outside of the Mach
Cone (r) (Fig. 50) . . ... .. .142
Cone Entirely Outside of the Cone (C) (Fig. 29) . 152
5.5 Supplementary Remarks on the Infinitely Flattened
Conical Flows . . ... 159

CHAPTER IV THE COMPOSITION OF CONICAL FLOWS AND ITS APPLICATION
TO THE AERODYNAMIC CALCULATION OF SUPERSONIC AIRCRAFT .. 168
4.1 Calculation of the Wings . .. 168
Symmetrical Problems . . ... ... 171
Rectangular Wings . .... ..... 171
Sweptback Wings . . ... 186
Lifting problems . . ... ..... 206
Rectangular Wings . . ... 214
Effect of Ailerons and Flaps . ... .225
Sweptback Wing . . ... .. 229
The Lifting Segments . .... ... 240
4.2 Study of Fuselages . ... ... 243
4.5 Conical Flows Past a Flat Dihedral. Fins and Control
Surfaces . . ... . 251
vi






NACA TM 1554


Page

REFERENCES .. .. .. . ...... 261

APPENDIX ................. ........... 264







































Digitized by the Interne[ Archive
in 2011 wilh funding Irom
University of Florida, Geolge A. Smathers Libraries wilh support from LYRASIS and the Sloan Foundation


Illp: www.arciiive.org details generaltheoryolc00unii






NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1354


GENERAL THEORY OF CONICAL FLOWS AND ITS APPLICATION

TO SUPERSONIC AERODYNAMICS*

By Paul Germain


CHAPTER I GENERALITIES ON CONICAL FLOWS

1.1 Equations of Supersonic Linearized Flows


1.1.1 General Equation for the Velocity Potential


Let us visualize the permanent irrotational flow of a compressible
perfect fluid for which the pressure p and the density p are mutual
functions. The space in which the flow takes place will be fixed by
three trirectangular axes Oxl, Ox2, Ox3, the coordinates of a fluid
molecule will be xl, x2, x3, the projections on Oxi of the veloc-
ity V and of the acceleration A of a molecule will be denoted by
ui and ai, respectively.


The fundamental equations which permit determination
are the Euler equations


of the flow


-* 1 -
A grad p
P



ai 1 6P
al P x,


the equation of continuity

*"La theorie ge6nrale des movements coniques et ses applications
a l'aerodynamique supersonique." Office National d'Itudes et de
Recherches Aeronautiques, no. 34, 1949.
1We employ the classic convention of the silent index: --pui
bxi v


is to be read: ~-(uI) + -~(Pu2)
JL d


S (Pu3)
6x






NACA U' 1354


div pV = 0


or x(Pui) = 0


(1.2)


and the equation of compressibility

p = f(p)


If one notes that


6ui
ai = u
1 k 6xk


and introduces the sonic velocity2


c2 dp
dp

the equation (I.1) assumes the form


p1 i
Sxi


_ 1 dp 6p
P dp 6xi


(1.3)


(1.4)


_ 2 ap
P 6xi


We introduce the velocity potential (xl, x2, x3), defined with


the exception of one constant, by


V = grad 4


x -
Ui ox,


2The velocity of sound, introduced here by the symbol d_ has a
dp
well-known physical significance; it is the velocity of propagation of
small disturbances. This significance frequently permits an intuitive
interpretation of certain results which we shall encounter later on
(see section 1.1.4).


oui
lk
axk


(1.5)






NACA TM 1354


which is legitimate since we shall assume the flow to be irrotational.
If we make the combination


u.u i N D 620
Uiuk uk- xi xk xi 0xk


one sees, taking into account equations (1.5) and (1.2), that


6' 6c 620 26 24
63 2 c2 62 (1.6)
3xi Oxk 6xi 6xk 6x 2
i


This equation is the general equation for the velocity potential.
One may show, besides, that c is a function of the velocity modulus;
thus one obtains an equation with partial derivatives of the second
order, linear with respect to the second derivatives, but not completely
linear.

The nonlinear character of the equation for the velocity potential
makes the rigorous investigation of compressible flows rather difficult,
at least in the three-dimensional case.

In order to be able to study, at least approximately, the behavior
of wings., fuselages, and other elements of aeronautical structures, at
velocities due to the compressibility, one has been led to introduce
simplifying hypothesis which permit "linearization" of the equation for
the velocity potential.


1.1.2 The Hypotheses of Linearization and Their Consequences

For aerodynamic calculation, one may assume that the body around
which the flow occurs has a position fixed in space nd that the fluid
at infinity upstream is moving with a velocity U, U being a constant
vector, the modulus of which will be taken as velocity unit. We shall
always assume that the axis Oxi has the same direction as U; the
hypotheses of linearization amount to assuming that at every point of
the fluid the velocity is reasonably equivalent to U.

We put in a more precise manner


S= 1+u u2 = v u3 = W






NACA TM 1354


u, v, w are, according to definition, the components of the "pertur-
bation velocity."

(1) u, v, w are quantities which are very small referred to
unity; if one considers these quantities as infinitesimals of the first
order, one makes it at least permissible to neglect3 in the equations
all infinitesimals of the second order such as u2, v2, uv, etc.

(2) All partial derivatives of u, v, w with respect to the
coordinates are equally infinitesimals at least of the first order so

that one is justified in neglecting terms such as u -, etc.
0x1 k fx2

One may deduce from these hypotheses a few immediate consequences:

(a) At every point of the field, the angle of the velocity vector
with the axis Oxi is an infinitesimal of the first order at least.
Hence there results a condition imposed on the body about which the flow
is to be investigated; at every point the tangent plane must make a
small angle with the direction of the nondisturbed flow (this is what
one calls the uniform motion, defined by the velocity U).

If one designates by q the velocity modulus, one has, taking the
hypotheses setup into account


q2 = (1 + u)2 + v2 + w2 = 1 + 2u

whence

q= 1 u


(b) The pressure p and the density p differ from the values p,
and pl which these magnitudes assume at infinity upstream only by an
infinitesimal of the first order; the equation (1.5) is written in effect


au C1
6xl P1 xl



3This signifies that u, v, w may very well not be infinitesimals
of the same order; in this case one takes as the principal infinitesimal
the perturbation velocity component which has the lowest order.






NACA TM 1354


with cl denoting the sonic velocity at infinity upstream; thus


2 -
u 1 (P pl)


5






(1'7)


On the other hand, according to equation (I.4)


P Pl = c2 P) = -lu


If one defines the pressure coefficient Cp by

SP P

p l /2 12


one has


Cp = -2u
p


(1.8)


(c) Finally, an examination of what becomes of the equation for the
velocity potential (equation (1.6)) under these hypotheses shows that it
is reduced to


x2

12


1 2
1 xl


+ _
x22


Let p(x1,x2,x3) be the "disturbance potential," that is, the
potential the gradient of which is identical with the disturbance-
velocity vector; P (x,x2,x3) is the solution of the equation with
partial derivatives of the second order


2
1 Cl2 _2
2 2
c1 6x1


_ i _
2
dX2


(1.9)


+ 32
Ox,2


a completely linear equation.


x \
2
3





NACA fM 1354


The Mach number of the flow is called the dimensionless con-

stant M which, with the velocity unit to be chosen arbitrarily,
cl
is written here M = 1/c1.


We put: (M2 1) = 02, with e being equal +1 or -1 according to
whether M is larger or smaller than unity.

(1) If M < 1, equation (I.9) is written


2 a29 2 29
P2 2 + 2p + 2 =
6x12 6x22 6x 2


an equation which may be easily reduced to the Laplace equation.

This equation applies to flows called "subsonic" because the velocity
of the nondisturbed flow is smaller than the sonic velocity at infinity
upstream. These flows will not be investigated in the course of this
report .

(2) If M >1, equation (1.9) is thus written


2 a2, -2, +629
032 + (I.10)
ax12 6x22 6x32


This equation applies to "supersonic" flows; if one interprets xl
as representing the time t, this equation is identical with the equa-
tion for cylindrical waves, well-known in mathematical physics. Investi-
gation of this equation will form the object of this report.

Remarks.

(1) It should be noted that, in order to write the preceding equa-
tion, it was not necessary to specify the form of the equation for the
state of the fluid. In particular, the formulas written above do not
introduce the value of the exponent 7 of the adiabatic relation p = kpy
which is the form usually assumed by the equation of compressibility.


Investigation of linear subsonic flows has formed the object of
numerous reports. See references 1 and 2.







NACA TM 1354


(2) The preceding analysis shows clearly the very different char-
acter of subsonic flows which lead to an elliptic equation, and of
supersonic flows which are represented by a hyperbolic equation.

(3) When we wrote equation (I.9), we supposed implicitly that
M2 1 was not infinitely small, that is, that the flow was not "tran-
sonic," according to the expression of Von KArman5. Thus it is impossible
to make M tend toward unity in the results we shall obtain, in the
hope to acquire information on the transonic case6.

(4) It may happen, in agreement with the statement made in foot-
note 3, that u is an infinitesimal of an order higher than first. In
this case, one will take up again the analysis made in paragraph (b) of
section 1.1.2, which leads to a formula yielding the Cp, more adequate
than the formula (1.8)


Cp = -2u (v2 + w2) (.11)


1.1.3 Validity of the Hypotheses of Linearization

Any simplifying hypothesis leads necessarily to results different
from those which one would obtain with a rigorous method. Nevertheless,
it was shown in certain numerical investigations on profiles (two-
dimensional flows) where the rigorous method and the method of lineari-
zation were applied simultaneously that the approximation method provided
a very good approximation for the calculation of forces. Besides, it
is well-known that the classic Prandtl equation for the investigation of


5Study of the transonic flows, with simplifying hypotheses analogous
to those that have been made, requires a more compact analysis of the
phenomena. It leads to a nonlinear equation, described for the first
time by Oswatitsch and Wieghart (ref. 3). From it one may very easily
deduce interesting relations of similitude for the transonic flows
(ref. 4). One may find these relations also, in a very simple manner,
by utilizing the hodograph plane.
In a general manner, according to the values of M, one may be led
to neglect certain terms in the final formulas found for the pressure
coefficient Cp. This requires an evaluation, in every particular case,
of the order of magnitude of the terms occurring in the formulas when M
varies. In this report, we shall never enter into such a discussion.
We shall limit ourselves voluntarily to the general formulas. An inter-
esting example of such a discussion may be found in the recent memorandum
of E. Laitone (ref. 5).






NACA TM 1354


wings of finite span in an incompressible fluid furnishes very acceptable
results, and the Prandtl equation results from a linearization of the
rigorous problem.

It happens frequently, we shall have occasion several times to point
it out, that the solution found for u, v, w will not satisfy the
hypotheses of section 1.1.2 in certain regions (for example in the neigh-
borhood of a leading edge); eventually certain ones among these magni-
tudes could even become infinite.

Under rigorous conditions such a solution should not be retained.
Anyhow, if the regions where the hypotheses of linearization are not
satisfied are "sufficiently small," it is permissible to assume that the
expressions found for the forces (obtained by integration of the pres-
sures) will still'remain valid. This constitutes a justification
a posteriori for the linearization method so frequently utilized in
numerous aerodynamic problems7. Therefore, we shall not systematically
discard the solutions found which will not wholly satisfy the hypotheses
we set up.


1.1.4 Limiting Conditions. Existence Theorem

Physically, the definition of sonic velocity leads to the rule
which has been called the "rule of forbidden signals" (see footnote 2
of section 1.1.1) and which can be stated as follows:

A disturbance in a uniform supersonic flow, of the velocity U
produced at a point P, takes effect only inside of a half-cone of
revolution of the axis U and of the apex half-angle a = Arc sin(l/M);
(D cot a) a is called the Mach angle, the half-cone in question Mach
after-cone at P."

Correlatively, one may state that the condition of the fluid at a
point M (pressure, velocity, etc.) depends only on the character of
the disturbances produced in the nondisturbed flow at points situated
inside of the "Mach fore-cone at M;" the Mach fore-cone at a point is
obviously the symmetrical counterpart of the Mach after-cone with respect
to its apex.

If one wants to justify this rule from the mathematical viewpoint,
one must start out from the formulas solving the problem of Cauchy and
take into account the boundary conditions particular to the problem.
Along the obstacle one must write that the velocity is tangent to the
obstacle which gives the value dP/dn. Moreover, at infinity

7
For instance, in the investigation of vibratory motions of infin-
itely small amplitude about slender profiles.






NACA TM 1354


upstream (xl = -c) the first derivatives of ( must be zero, since 9
is, from the aerodynamic viewpoint, only determined to within a constant,
it will be assumed zero.

The characteristic surfaces of the equation (I.10) are the Mach
cones. If one of the Mach cones of the point P cuts off a region (R)
on a surface (Z), the classic study of the problem of Cauchy8 shows
that the value of 9 at P is a continuous linear function of the
values of 9 and of dcp/dn on R.

Let us therefore consider a point M of a supersonic flow such
that its fore-cone does not intersect the obstacle. We take as the
surface E a plane xl = -A, with A being of arbitrary magnitude.
On E, rP and dP/dn, which are continuous functions, will be arbi-
trarily small. Consequently the value of c at M is zero. Thus one
aspect of the rule of "forbidden signal" is justified.

Let ds suppose that the forward-cone of M cuts off a region r(M)
on the obstacle; on r(M), dc/dn is given by the boundary conditions;
thus 91(M) is a linear function of the values of C on r(M).

One sees therefore that, if one makes M tend toward a point MO
of the obstacle, one will obtain a functional equation permitting the
determination of P on the obstacle, at least in the case where the
existence and uniqueness of the solution will be insured9. Consequently,
((M) depends only on the values of CP/dn in the region r(M); this
justifies the fundamental result of the rule of "forbidden signals."10


1.1.5 General Methods for Investigation

of Linearized Supersonic Flows

In a recent articlell dealing with the study of linear supersonic
flows, Von Kirman indicates that two major general procedures exist for

8For the problem of Cauchy, relative to the equation for cylindrical
waves, see for instance references 6 and 7.
9Such a method has been utilized by G. Temple and H. A. Jahn, in
their study of a partial differential equation with two variables (ref. 8).
10A more exact investigation of this question may be found in
appendix 1, at the end of this report.
11See reference 4. A quick expose of the methods in question may
also be found in the text, in reference 2.






NACA TM 1354


the study of these flows, one called "the source method," the other
"the acoustic analogy."

The first is an old method and its theoretical application is
fairly simple. It consists in placing on the outer surface of the
obstacle a continuous distribution of singularities, called sources,
the superposition of which gives at every point of the space the desired
potential; the local strength of the sources may, in general, easily be
determined with the aid of the boundary conditions. The second method
utilizes a fundamental solution of the equation (I.10), the composition
of which permits one to obtain the desired potential; this procedure is
interesting in that it permits utilization of the Fourier integrals and
thus furnishes, at least in certain particular cases, rather simple
expressions for the total energy.

Von Karman also indicates, at the end of his report, a third general
procedure, that of conical flows.

We intend to investigate in this report the conical flows and
the development of this third procedure which utilizes systematically
the composition of the "conical flows" and, more generally, of the flows
which we shall call "homogeneous flows of the order n." We shall see
that this procedure permits one to find very easily, and frequently
with less expenditure, a great number of the results previously obtained
by other methods, and to bring to a successful end the investigation of
certain problems which, to our knowledge, have not yet been solved.


1.2 Generalities on Conical Flows

1.2.1 History and Definition

Conical flows have been introduced by A. Busemann (ref. 9) who has
given the principal characteristics of these flows and has indicated
briefly in what ways they could be utilized in the investigation of
supersonic flows. Busemann gives as examples some results, frequently
without proof. Several authors have supplemented the investigation of
Busemann: Stewart (ref. 10) has studied the case of the lifting wing A
to which we shall come back later on; L. Beschkine (ref. 11) has fur-
nished a certain number of results but generally without demonstration.
We thought it of interest to attempt a summary of the entire problem.

One calls "conical flows" (more precisely, "infinitesimal conical
flows")12 the flows in which there exists a point 0 such that along

12The adjective "infinitesimal" is remindful of the fact that the
flows have been linearizedd;" we shall henceforward omit this qualifica-
tion since no confusion can arise in this report.






NACA TM 1354


every straight line issuing toward one side of 0, the velocity vector
remains of the same value.

Let (i) be a plane not containing 0, normal to the vector U;
let us suppose only that the velocity vector at every point of (i) is
not normal to (r); the projection of these velocity vectors on (i)
determines a field of vectors, the lines of force of which we shall
call (y): the cones (a) of vertex 0 and directrix (7) are "stream
cones" for the flow.

More generally, let (S) be a stream surface of the flow, passing
through O; every surface deduced from (S) by homothety of the center 0
and of k, k being an arbitrary positive number, is a stream surface.
(S) is not necessarily a conical surface of apex 0, but having (S)
given as an obstacle does not permit one to foresee the existence of
such a flow. It is different if a conical obstacle of apex 0 is given;
the designation "conical flow" is thus justified.

Conversely, let us consider a cone of the apex 0, situated entirely
in the region xl >0, and suppose that a linearized supersonic flow
exists around this cone; this flow is necessarily a conical flow such
as has just been defined; in fact, if V(xlX2,x3) denotes this velocity
field, V( xl,Xx2,Xxj) ( being any arbitrary positive number) is
equally a velocity field satisfying all conditions of the problem; con-
sequently, if the uniqueness of the desired flow is admitted, 7 must
be constant along every half-straight line from 0 13.

Let us also point out that according to equations (1.8) or (I.ll),
the surfaces of equal pressure are also cones of the apex 0.


1.2.2 Partial Differential Equations Satisfied

by the Velocity Components

According to definition, the velocity components of a conical flow
depend only on two variables; on the other hand, as functions of xl,


13It should be noted that this argument will no longer be valid
without restriction in the case of a real supersonic flow around a cone
because in this case the principle of "forbidden signals" is no longer
valid in the rigorous form stated. Among other possibilities, a detached
shock wave may form upstream from the cone behind which the motion is
no longer irrotational.






NACA TM 1354


x2, x3, they are naturally the solution of the equation


S2f
6x3 2
O3


Let us first put


x2 = r cos 9


x3 = r sin 0


the equation then assumes the form


2 32f
6xl2


2 2
_ 2f r1 2 1 rf
or2 r2 602 r 3r


(1.12)


The second term of equation (1.12) is actually nothing else but
the Laplacian of f(x1,x2,x3) in the plane x2, x3 (xl being con-
sidered as parameter); naturally f(xl,r,8) is periodic in 8, the
period being equal to 2n.

To make the conical character of the flow evident, let us put


xI = prX


(I.13)


X is a new variable; X < 1 characterizes the exterior of the Mach
cone with the apex 0, X > 1 characterizes the interior of the cone.
Under these conditions, the disturbance-velocity components are func-
tions only of :. and e. Since f is a function of X and 0 only


d2f = 2 dx2
dx2


+ 232 dv dB +
ax 6e


02f
O82


+ 6f d2X + kf d2
5ax e


dX = r (dxl OX dr)
o~r\ 1I


d2x = -L(d2x X d2r -
pr


2dr dx 1 + 20 X dr2)
r r


2 62f
6x 2


_ 2f
bx22






NACA TM 1354


2f
;r2


62f
6e2


6f are the respective coefficients of dxl2,
Tr


dr2,


d2r in the expression of d2f as a function of the vari-
s xl, r, e.

As a consequence, the equation (1.12) becomes under these conditions


r(x2
6-A.O2


(1.14)


+ b2f + f = 0
+02 -
6e2 ax


One may try to simplify this equation further by replacing the
variable X by the variable t, X and t being connected by a rela-
tionship X = X()), and by making a judicious choice for the func-
tion x(S). The first operation gives


(x2 2 i)
6t2


+ y,2 r
602


+ B 0
6 at (x. )x' =
oS X'


with the primes denoting derivatives with respect to E.
this equation, one may make the term in disappear.
realized by putting

(1) If X >1,


For simplifying
This will be


X = ch e


one obtains for f Laplace's equation


62f
6t2


+ 2
S62


(1.16)


(2) If X < 1,

= cos 1 (I.17)

in this case, one obtains the equation for waves with two variables


62f
3,2


62f
682


(I.18)


(1.15)






NACA TM 1354


Geometrical interpretation.- X > 1 corresponds to the interior of
the Mach rearward cone (r) of the point 0; every semi-infinite line,
issuing from 0, inside of this cone, has as image a point 8, E. One
will assume, for instance, -n < 0 < i; k = 0 corresponds to the
cone (r), E = w corresponds to the cone axis (it will always be pos-
sible to assume as positive). The image of the interior of (r)
forms therefore on the region (A) of the plane (0,a) (fig. 1), limited
by the semi-infinite lines AT, A'T' and by the segment AA'. The
correspondence is double valued in the sense that to a semi-infinite
line issuing from 0 there corresponds one point and one only (0e,)
in the bounded region and conversely, to one point of this region there
corresponds one semi-infinite line, and one only, issuing from O,
inside of (r).

Since we shall suppose, in general, that the cone investigated is
entirely in the region xl > 0, only this region will be of interest
(P then being identically zero for xl < 0). The semi-infinite lines
of this region issuing from 0, outside of (r), correspond to 0 < X < 1
(fig. 2), that is, according to equation (I.17), 0 < r < !L; n = 0
2
corresponds to the cone (F), T = n to the plane xI = 0; the semi-
2
infinite lines issuing from 0 correspond biunivocally to the points
of the region (A'), inside of the rectangle AA'B'B in the plane (e,r).

Summing up, the velocity components satisfy the simple equa-
tions (I.16) and (I.18), the first of which is relative to the region (A),
the second to the region (A').


1.2.3 Fundamental Theorem

The equation (1.14) which represents the fundamental equation of
our problem is an equation of mixed type; it is elliptic or hyperbolic
according to whether is larger or smaller than unity. In order to
study this equation in a simpler manner, we have been led to divide the
domain of the variables into two parts and to represent them on two
different planes. How an agreement will be reached between the solutions
obtained for f in the two planes that is the question which will be
completely elucidated by the following theorem which will be fundamental
in the course of our investigation.

Theorem: There exists "agreement" as to X = 1 for all derivatives of

f, defined in either the region (A) or (A'), provided that there is


"agreement" for the function itself.






NACA TM 1354


In fact, let us take two functions fl(e,), f2(6,Ti), the first
satisfying the equation (1.16) in the region (A), the second the equa-
tion (I.18) in the region (A'), both assuming the same values c(e) on
the respective segments ( 0 = 0, -i < 0 < n) (r = 0, -n < 0 < i). Let
Inf
B0 be the abcissa of a point of AA'. If 6nfL(80,0) exists,
8en


Sd-; consequently
den


nf 2 (0,0)
en


exists and


ae oO
aen


a3f(e,, )
- On


Let us now pass to the investigation of the derivatives of the
order n of the form nf the equation (1.14) shows first that
6Xoen-1


f(e,1) =- -f(,1)
OX 602


which proves that all partial derivatives of the order 1 with respect
to X have the same value on (P), whether they are calculated starting
from fl or from f2. The argument develops without difficulty through
recurrence. By deriving equation (1.14) n times with respect to :
and making X = 1, one obtains


(2n + 1) + n2 + -6nf 0
An+1 .a 2" 2n



which finally shows that the values n+f can be uniquely expressed
ae. n
as a function of the derivatives of f(0) with respect to G and that
they, consequently, have the same value, whether calculated starting
from fl or from f2.

Summing up, one may say that it is sufficient for the establishment
of the "agreement" between two solutions defined in (A) and (A'), if
these solutions assume the same value on the segment AA'.


anfl(00
agn






16 NACA TM 1354


1.2.4 Mode of Dependence of the Semi-Infinite

Lines Issuing From 0

If one puts in the plane (e0,)

8 + T = 2X 8 q = 2 (1.19)

one sees that the characteristics of the equation (I.18) are the parallels
to the bisectrices K = cte, = cte. These characteristics are, in
the plane (1,e), the images of the planes

xI = -r cos(2A 0) and xl = or cos(O 2P)


which are the planes tangent to the cone (P). The characteristics
passing through a point s0(O0,0) are the images of two planes tangent
to the cone (r) which one may lay through the semi-infinite A0 cor-
responding to the point 60 of the plane (8,0) (fig. 3). The gener-
atrices of contact are characterized on the cone by the values 81
and 82 of the angle 0. One encounters here a result which seems to
contradict indications of section 1.1.4. This apparent contradiction
is immediately explained if one notes that, since all points of a semi-
infinite A0 issued from 0 are equivalent, one must consider at the
same time all Mach cones, the apexes of which are situated on A0; the
group of these cones admits as envelope precisely the two planes tangent
to the cone (r) passing through A0. We shall call "Mach dihedron
posterior" to the semi-infinite A0 that one of the dihedra formed by
the two planes which contains the group of the Mach cones to the rear of
the points of A0. The region inside ot this dihedron and outside of
the cone (r) has as image in the plane (0,rj) the triangle 01 5062.
A semi-infinite A1 will be said to be dependent on or independent of
A0 according to whether the image of A1 will be inside or outside of
the triangle 81 6082. This argument also explains why the equa-
tion (1.14) shows elliptic character inside of (r). More precisely,
two semi-infinite lines A1 and 62, inside of (r), are in a state of
neutral dependence (ref. 9). In fact, let M1 be a point of Ak, M2
a point of 2; let us suppose that Mi is outside of the Mach forward
cone of M2; according to the argument of section 1.1.4 the point M2
seems to be independent of Ml; but on the other hand, if one assumes Mi'






NACA TM 1354


to be a point of A1, inside of the Mach forward cone of M2, M1'
and M1 are equivalent which explains that M2 is actually not inde-
pendent of M1 (fig. 4).


1.2.5 The Conditions of Compatibility

Thus one may foresee how the solution of a problem of conical flow
will unfold itself. One will attempt to solve this problem in the
region (A') which will generally be fairly easy since the general solu-
tion of the equation (1.18) is written immediately by adjoining an arbi-
trary function of the variable 6 + n to an arbitrary function of the
variable 6 q. This will have the effect of "transporting" onto the
segment AA' the boundary conditions relative to the region (A').
Applying the fundamental theorem, one will be led to a problem of har-
monic functions in the region (A). But taking as unknown functions the
components u, v, w, of the disturbance velocity, we have introduced
three unknown functions (while there was only one when we dealt with
the function C). One must therefore write certain relationships of
compatibility which express finally that the motion is indeed irrotational.

The motion will be irrotational if u dxI + v dx2 + w dx3 is an
exact differential which will be the case when, and only when

x1 du + x2 dv + x dw = r(py du + cos 8 dv + sin 0 dw)


is an exact differential. This can occur only if this expression
is identically zero, with u, v, w being functions uniquely of 0
and of X (the total differential not containing a term.in dr must
be independent of r):

In a conical flow the potential is written


P = uxl + vx2 + wx3 = r(puX + v cos 0 + w sin e)


with u, v, w being the disturbance-velocity components.

One will note that C, is proportional to r.

Moreover


OX du + cos 0 dv + sin 0 dw = 0


(1.20)






18 NACA TM 1354


This is the relationship which is to be written, and this is the
point in question, on one hand in the plane (C,0), on the other in the
plane (6,).

(a) Relations in the plane (0,q). One may write

u = ul(?) + u2(p)


and analogous formulas for v and w, X and
relations (I.19). One has in particular

dUl 3u + u du2 u
dX 5q 60 du o0

Besides, according to equation (1.20)


- being defined by the


P cos T dul + cos 0 dvl + sin 0 dwl = 0


(1.21)


3 cos T du2 + cos 0 dv2 + sin 0 dw2 = 0


however: 0 = x + P, T = A 1; and consequently the first equa-
tion (1.21) is written


cos r 1 cos K dui + cos K dvI + sin k dwli +


sin psin du sin K dv1 + cos K dw = 0


since the two quantities between brackets are unique functions of
the preceding equality causes

p cos K dul + cos K dvi + sin X dwl = 0

P sin k dul sin X dv1 + cos K dwI = 0


dv1
-p dul = 1
cos 2X


dwl
sin 2X


(1.22)






NACA TM 1354


In the same manner one will show that


dv2 dw2
-p du2 = --
cos 2Ii sin 21


(1.23)


(b) Relations in the plane (e,t).

The calculation is perfectly analogous. The equation (1.16) causes
us to introduce the complex variable 0 = 8 + it and the func-
tions U(.), V(C), W(t), defined with the exception of an imaginary
additive constant, the real parts of which in (A) are, respectively,
identical to u(e,a), v(0,t), w(e,t).

The equation'(I.20) permits one to write

0 ch t dU + cos 0 dV + sin 0 dW = 0


If one puts


e + it =


e iE5 =


one obtains


cos T L
2sin


sin [


cos dU + cos -
2 2


sin dU sin -
2 2


dV + sin dW +
2


dV + cos -
2


thence one concludes as previously


- dU dV dW
cos ( sin c


The formulas (I.22), (1.23), (I.24) express the relationships of
compatibility which we had in mind.


dW = 0


(1.24)






NACA TM 1354


Remark.

We shall utilize frequently the conformal representation for studying
the problems relative to the domain (A). If one puts, in particular


Z = eit = e- ei


one sees that (A) becomes in the plane Z the interior area of the
circle (CO) with the center 0 14 and the radius 1 (fig. 5).


If one puts Z = pe the point Z is the image of a semi-infinite
line, issuing from the origin of the space (xl,x2,x3), characterized
by the angle 6 and the relationship


X1_ 1 + p2
pr 2p


The origin of the plane Z corresponds to the axis of the cone (P),
the circle (Co) to the cone (r) itself. A problem of conical flow
appears in a more intuitive manner in the plane Z than in the plane f.
In the plane Z, the formulas (1.24) are written


SdU = 2Z dV = 2iZ dW (1.25)
Z2 + 1 Z2 1


We shall moreover utilize the plane z defined by


S 2Z
Z2 + 1


The domain (A) corresponds conformably to the plane z notched by
the semi-infinite lines Ax, A'x' (fig. 6), the cone (r) at the edges
of the cuts thus determined, and the axis of the cone (F) at the origin



14
No confusion is possible between the point 0, origin of the sys-
tem of axes xl, x2, x3 and the point 0, here introduced as the
origin of the plane Z.






NACA TM 1354


of the plane z. The relations of compatibility in the plane z then
assume the form


-0 dU = z dV iz dW (1.26)
1 -z2


1.2.6 Boundary Conditions

The Two Main Types of Conical Flows

The boundary conditions are obtained by writing that the velocity
vector is tangent to the cone obstacle. Let, for instance, x2(t), x3(t)
be a parametric representation of the section xI = 0 of the cone;
x3x2' x2x'3, xx3', -PY2' constitute a system of direction parameters
of the normal to the cone obstacle, and the boundary condition reads


x' vx3' = (x3x2' x2x3' (1 + u) (1.27)


It will be possible to simplify this condition according to the
cases. However, the simplification will have to be treated in a dif-
ferent manner according to the conical flows investigated. As set forth
in section 1.1.2, two main types of conical flows may exist.

(1) The flow about cones with infinitesimal cone angles, that is,
cones where every generatrix forms with the vector U an angle which
remains small. Naturally, the cone section may, under these conditions,
be of any arbitrary form; since the flow outside of (P) is undisturbed
(velocity equivalent to U), on the cone (r) u, v, w are zero.

The problem may have to be treated in the plane Z; U(Z), V(Z),
W(Z) will have real parts of zero on (CO). The image (C) of the
obstacle, in the plane Z, is defined by a relation p = f(e); conse-
quently, a parametric representation of the section xi = p will be
obtained by means of the formulas


2p 2p
X2 = 2 cos e X3 sin G
1 + p2 1 + p2






NACA TM 1354


Thus the condition (1.27) becomes


w sin 0 p' cos a + p2(p sin e + p' cos 8) +

v cos 0 + P' sin e + 2(p cos 0 p' sin 8) = 2-(1 + u) (1.28)


with 0 taken as parameter, and p' denoting the derivative of p
with respect to 0. The investigation of conical flows with infinitesimal
cone angles will form the object of chapter II.

(2) The flow about flattened cones, that is, cones, the generatrices
of which deviate only little from a plane containing U. Let us remember
that (section 1.1-.2) the tangent plane is to form a small angle with 6;
consequently, rigorously speaking, the section of such a cone cannot be a
regular closed curve, an ellipse for instance; it must present a lentic-
ular profile (fig. 7). In chapter III we shall study the flows about
such cones.

Remark.

Actually, we have, therewith, not exhausted all types of conical
flows, that is, those for which linearization is legitimate. One may,
for instance, obtain flows about cones, the section of which presents
the form shown in fire 8; the axis of such a cone has infinitely small
inclination toward U.

Before beginning the study of these flows we shall, in order to
terminate these generalities, introduce a generalization of the flows,
the possible utilization of which we shall see in a final chapter.


1.3 Homogeneous Flows

1.3.1 Definition and Properties

The conical flows are flows for which the velocity potential is of
the form


cP = rf(e,X)

as we gad seen in section 1.2.5. One may visualize flows for which


P = rnf(0,x)






NACA TM 1354


We shall call them homogeneous flows of the nth orderl5. The conical
flows defined in section 1.2 are, therefore, homogeneous flows of the
order I. However, we shall maintain the expression "conical flow" to
designate these flows since this term has been used by numerous authors
and gives a good picture.

The derivatives of the velocity potential with respect to the vari-
ables xl, x2, x3 all satisfy the equation (I.10). If one then con-
siders the derivatives of the nth order of the potential of an homogeneous
flow of the nth order, one finds that they depend only on X. and a
and satisfy the equation (1.14); the analysis made in section 1.2.2
remains entirely valid. One may make the changes in variables (I.15)
and (T.17) which lead to the equations (1.16) and (1.18). Thus one has
here a method sufficiently general to obtain solutions of the equa-
tion (I.10) which may prove useful.

The simplest flows are the homogeneous flows of the order 0 which
do not give rise to any particular condition of compatibility. For the
flows of nth order, in contrast, one has to write a certain number of
conditions connecting the derivatives of nth order. We shall examine16
as an example the case of homogeneous flows of 2nd order.

There are six second derivatives which we shall denote Tij (i
and j may assume independently the values 1, 2, 3), Cij designating

4 Outside of (r) we shall put
6xi 6xj



ij ij i

1 2
with P.ij being a function of X only, Pij2 of p only (see for-
mula 1.19). Inside of (r), ij. is the real part .of a function .i.(U).

In order to obtain the desired relations, it is sufficient to note
that


15The definition for homogeneous flows of the nth order has been
given for the first time by L. Beshkine (ref. 11); this author, by the
way, calls them conical flows of the nth order. One may also connect
this question with the article of Hayes (ref. 12).
S6ee appendix 2.






NACA TM 1354


ij dxj = dPi


and to apply the results of section 1.2.5; thus one may write the fol-
lowing six relations between the c 1


-0 dil 1 d 1 d 1 (i = 1,2,3)
11 cos 2X 12 sin 2X i3

which, besides, are reduced to five as one sees immediately. One will
have analogous relations for the functions Cpj2 (it is sufficient to
exchange the role of X and of p).

Finally, one has for the analytic functions oij()


d 1 d 1 d3
ii cos i i2 sin i i3

namely six relations which as before are reduced to five. The written
conditions are not only necessary but also sufficient since the func-
tions Pi necessarily are the components of a gradient. Thus one sees
that there is no difficulty in writing the conditions of compatibility
for a homogeneous flow of nth order.


1.3.2 Relations Between the Homogeneous Flows

of nth and of (n-l)th Order

We shall establish a theorem which can be useful in certain prob-
lems and which specifies the relations existing between homogeneous
flows of nth and of (n-1)th order; we shall examine the case where n = 1.

1.3.2.1.- Let us consider inside of the cone (P) a homogeneous
flow of the order 0 defined by


f = REcZ)]






NACA TM 1354 2


We shall first of all seek the components u, v, w of the dis-
turbance velocity


dC = u dxI + v dx2 +


wdx = R[O'(Z) dz = R[Zs'(Z) d


then


1 + p2
xl = pr 2
2P


thus


dZ = dp + i d0
Z p


x2 = r cos 0


x3 = r sin 8


_ 2 + 1 dx x2 dx2 + x2 dx
p2 1L r2


Sx2 dx3 x3 dx2
r2


whence one deduces


p2 + 1 1
p2 1 1-


v cos P02 + 1
r P2 1


w sin 8 02 + 1
r p2 1


R[Zq'(Zz] + sin e T[ Z'(z]



R-Zo'(Z CS- '(Z
~r -


p2 + 1 sin 0
P

p2-1 sin 0
P


however


Z 1
z


Z
z


p2 + 1

P21
_ 2 -
p


cos 8 + i



cos e + i






NACA 4M 1354


hence the result


1
u



xl



xl


p2 + 1 R Z'(Z)
p2


p2 +1




p2 -1
p2 1


R [- (Z2 + 1)'(Z)


R (2 1)0,(Z)]
R- 2 1)01(z


(1.29)


1.3.2.2.- Let us now consider a point O' (xI = E1, x2 = 0, x3 = 0),
El being a very small quantity. Let M be a point with the coordi-
nates (xl,r,e) with respect to O, inside of (F), and with the para-
meters (p,8) in the plane Z. For the conical flow (homogeneous of
Ist order) with the vertex 0', its coordinates are: (xl- E r, a) and
its parameters in the Z-plane:


( p2 1 1


since


dxI = 1 = Br


p2 1
2p2


dp = x 2 1 dp
do = x p
p + 1


Let us then consider two identical conical fields but with the
apexes 0 and 0', and form their difference. We shall obtain a
velocity field which, due to the linear character of the equation (I.10),
will satisfy this equation. If

U0 = R[(Z)


denotes the component u of the field with the vertex
component u in the "difference field"


0, one has as


u = +RF(Z) R 2 + 1 z
P2 X1 ]


- P2 + 1 1 RFZF'(Z)
p2 1 xl
~~2


(I.30)





NACA TM 1354


El being considered as infinitely small. Moreover, according to the
relations (1.25), the components v and w are written


v = 2 + 1 aE fi (Z2 + 1)F'(Z]

w l p2 +- i PZ2 1)F'(Z)

xi p2 1 2


1.3.2.3.- Let us consider the point 0''(O,E2,0), with E2 being
a small quantity.' Let M be a point with the coordinates (x1,r,O)
with respect to 0, inside of (r), with the parameters (p,8) in the
plane Z. For the flow with apex 0'', the coordinates of M are
(x1, r E cos 8, 0 + E2 sin as can be easily stated by projecting
M in m on the plane x2x3 (fig. 9). But on the other hand


2xd 1 2
dr dp = -C2 cos 0
( (1+p2)2


r de x P dO = 2 sin 0
1 + p2

thus


dZ = e p + ip d] = e2 12 2 x i sin e cos 8 + e1i


with Z + dZ representing the point M in the conical field with the
vertex 0''.

Let us then consider two identical conical flows, but with the
apexes 0 and 0'', and form their difference. We shall obtain a
velocity field which due to the linear character of the equation (I.10)
will satisfy this equation. If

O = R[G(Z)






28 NACA TM 1354

denotes the component v in the field of the vertex 0, one has a com-
ponent v in the "difference field"

v +n[G(Z) R[(Z + dZ) = -R[G'(Z)dZ]

Cx2 P2 +_1 2) + i sin 1(P2
= p-- R G'(Z) cos e ( + p2) + i sin e(p2 l)e


S2 P 2+ 1 R ZG'(Z) Z +1
2x1 2 -1 2
2x p2 1-L Z/j


x1 P2 1 2 (1.32)


besides, according to equation (I.25), the components u and w are
written


E2 02 + 1
1 po' 1
(1.33)
w =2 p2 + 1 R (Z2 1)G'(Z]
x p2 1 2


1.3.2.4.- With these three lemmas established, it is easy to demon-
strate the property we have in mind. Let us call "complex potential" of
a homogeneous flow of zero order the function ((Z) (section 1.3.2.1)
so that

P = R [0(Z)]

so that the function of complex variable, the real part of which gives
insideof (P) the projection of the disturbance velocity in the direc-_
tion i, is the "complex velocity" of a conical field in the direction 1;
so that, finally, the velocity field obtained by the difference of two iden-
tical conical fields, the verties of which are infinitely close and
ranged on a line parallel to 1, is the "field derived from a conical
flow" in the direction 1; then we may state:






NACA TM 1354


Theorem: The field derived from a conical flow in the direction Z is

the velocity field of a homogeneous flow of zero order; the complex

potential of that flow of zero order is proportional to the complex

velocity of the conical field given in the direction i, since the pro-

portionality factor is real.

The proof follows immediately. According to sections 1.1.2
and 1.1.3 one may be satisfied with considering, for definition of a
homogeneous flow, the inside of the cone (r); comparison of the for-
mulas (1.29), (I.30), (I.31), (1.32), (1.33) entails the validity of
the above theorem'if I is parallel or orthogonal to U. Hence the
general case where I is arbitrary may be deduced immediately; if
F(Z), G(Z), H(Z) are the complex velocities in projection on Oxl,
Ox2, Ox3, the expression for the component u of the field derived in
-4
the direction I(EE 2,3) is



u = p2 + 1 R Z F'(Z) + e20'(Z) + 3H'(Z)
u i 02 1 C


Thus, with E1F(Z) + E2G(Z) + E3H(Z) being the complex velocity in
projection on 1, comparison of this formula with the first formula (I.29)
completely demonstrates the theorem.
-4
Corollary: The field derived in the direction 2 of a conical flow,

the complex velocity of which in the direction I is K(Z), is a

velocity field of a homogeneous flow dependent only on K(Z) (not on

the direction I).

The theorem just demonstrated may be extended without difficulty
to the homogeneous flows of nth and (n-l)th order. A statement of this
general theorem would require only specification of a few definitions;
however, since we shall not have to utilize it later on, we shall not
formulate this statement.






NACA TM 1354


CHAPTER II CONICAL FLOWS WITH INFINITESIMAL CONE ANGLES *

2.1 Solution of the Problem

2.1.1 Generalities


We shall now treat the first problem set up in section 1.2.6. We
shall operate in the plane Z. Let us recall that the image of the
cone (r) is the circle (CO) of radius unity centered at the origin,
and that the image of the obstacle is a curve (C), defined by its polar
equation p(e). We shall denote by (D) the annular domain comprised
between (C) and (CO); we shall call (O0) the circle of smallest
radius centered at the origin and containing (A) in its interior, and
we shall call k the radius of the circle (70). In this entire
chapter, k will be considered as the principal infinitesimal.

The problem then consists in finding three functions U(Z), V(Z),
W(Z) defined inside of (D) except for an additive imaginary constant,
so that

(1)
-3 dU 2Z dV 2iZ dW (1.25)
Z2 + 1 Z2 1

(2) the real parts u, v, w, which are uniform become zero on (CO),

(3) on (C), one has the relation


vp cos e + p' sin 8 + p2(p cos e p' sin e) +

r -j 2
p sin e p' cos e + p2( sin 0 + p' cos e) = 2-2( + u)


Put in this manner, the problem is obviously very hard to solve in
its whole generality; however, an analysis of the permissible approxima-
tions will simplify it considerably.


2.1.2 Investigation of the Functions U(Z), V(Z), W(Z)

2.1.2.1.- An analytical function of Z will be the said func-
tion (A) if its real part becomes zero on (Co). Let us designate by
NACA editor's note: Some minor inconsistencies appear in the number
of equations in this chapter and subsequently in chapters III and IV, but
attempt was made to change the numbering as given in the original text.






NACA TM 1354


(70') the circle with the radius 1/k, centered at the origin, and by
(D') the annulus limited by (o0) and (70') (fig. 10).

Lemma I.- A uniform function (A), defined inside the annulus

limited by (70) and (CO) may be continued over the entire domain (D').

This results immediately from Schwartz' principle. Let M and M'
be two symmetrical points with respect to (Co), M being inside of
(CO); 6ne defines the function (A) at the point (M') as having,
respectively, an opposite real and an equal imaginary part compared to
the real and the imaginary part of the function given at the point M.

Lemma II.- A holomorphic function (A) inside of (D') has a

Laurent development of the form17


+ 2IQnn~izn
ip+ Z Kn)
1

Let H(Z) = h + ih' be such a function (A). Let us write its
Laurent development in (D') provisorily in the form


H(Z) = JnZn + -n
0 1

It is an immediate demonstration and yields the formulas defining Jn
and Kn


Kn = (h + ih')o7eind8
21t 0O


17
We remember that Kn denotes the conjugate imaginary of Kn.






NACA TM 1354


(h + ih')70 denoting the value of H on (70); likewise


J -kn 2
I230


(h + ih') ie-inede
(h~ih '70


Consequently, according to the lemma I:


Kn = -Jn


moreover


S2n
H(Z) d
Z 2xJo


S.2n
(h + ih')0 dO -
00 2n 0o


is purely imaginary, and the lemma II is therewith demonstrated.


We shall note that, if H(Z) is limited by M on (70)
one has the inequality


or (70'),


Kn < Mkn (II.]

Lemma III.- A function (A) with a real and uniform part defined

in (D) can be developed inside of (D') in the form


B log Z + ip + (f KnZn (II._



with B being real.


Actually, the derivative of the function (A) is necessarily uni-
form. Thus one knows (see for instance ref. 13) that one may consider
the given function as the sum of a uniform function H(Z) and a loga-
rithmic term; since the critical point of the logarithm is arbitrary
inside of (O), it is particularly indicated to choose this point at
the origin; since the real part of the function is uniform, the coeffi-
cient of log Z is real. Besides, since log Z has a real part zero


h d
Co


Jo i-
2i ,0
co






NACA TM 1354


on (Co), H(Z) is itself a function (A). The given function may
therefore be continued inside of (D') and the development (11.2) is
thus justified.

Remark.


(7o)
form


If one chooses as pole of the logarithmic term a point inside of
but different from the origin, one obtains a development of the


B' log aa -- + i8 +
1 1


- K'nZn


2.1.2.2.- The functions U,
three functions (A) with a real
be developed in the form (II.2).


- e u(z)
2


= A log Z + ia


V(Z) = B log Z + ip



W(Z) = C log Z iy


+
ixz


V, W of the variable Z are all
uniform part and, consequently, can
We shall write henceforward


(n JZ
Znn nZn)


J K n Zn


- n


- L,'n
Lnzn)


(11.3)


A, B, C are real, a, 0, ? are real and also arbitrary; but these
developments are not independent since the relations (I.25) must be
taken into account. For instance, Z dV/dZ must be divisible by Z2 + 1;
otherwise we would have for U logarithmic singularities on the cone (r)
which is inadmissible. Now


=B-i
1


n(K"n + izZn
Zn|- + K, I


zd
dZ


(+ Ln
1





NACA TM 1354


Hence one deduces the relations


B 1= ( P[K2p + R2]
1


S=3 ( 1)P(p + 1)K2p
0


- Kp+1


obtained by putting in the preceding equality Z = i and


Z = -i.


Z dW/dZ must be divisible by Z2 1 which gives


C = 2p (L2p + L2p)
1


0 => (2p + 1)(L2p+1 + L2p+l)
0


Finally, the qualities (1.25) lead, in addition,
connecting the coefficients of the developments (II.3)
thus one may write the relations


B + 2K2 =-i[C 2L2]


nKn (n 2)Kn2 =


to relationships
among themselves;


K1 K1 = -i + L]


i[(n 2)Ln-2 + nr]


(n >,2)


and on the other hand


B =-(J + J1

K1 = -A + 2J2


nKn = (n 1)Jn-1 +


Likewise,


(n >2)


(n + 1)Jn+1






NACA TM 1354


2.1.2.3. Approximations for the developments (II.3).- Moreover,
the hypotheses of linearization must be taken into account which, as we
shall see, will permit us to simplify the developments (11.3) consider-
ably and will lead us in a very simple manner to the solution of the
problem posed in section 2.1.1.


The qualities (11.6) make V(Z) and W(Z)
order. We shall denote by M an upper limit of
circle (70). M will be equally an upper limit

(70') and hence in the entire domain (D').


seem of the same
their modulus on the
of their modulus on


If one utilizes the inequality (1.1), (11.4) shows that18


B = 0 Mk2)


K1 1 = o(Mk2


If one assumes a, 0, 7 zero in what follows, which does not at
all impair the generality, one may write the second formula (11.3) in
the form


V(Z) R(Kn ( -



and consequently:


2
2


Kn = B log Z -
Zn


In the annulus limited by (70) and
equality is

o(Mk2log k)

Likewise according to equation (11.5)


C = o(Mk2)


(Co), the second term of this


L1 + i =o(Mk3)


W(Z) iT(L1)(. + Z) n = C log Z + RL)
2


- z)


- n
2


180 denotes Landau's symbol, A = 0 Mk2) signifies that A is
Mk2
limited when k tends toward zero.


nZn + iT(K1)(


+ z)





NACA TM 1354


In the annulus comprised between (70) and (CO), the second term
of this equality is also

S(Mk21og k)

Furthermore, according to equation (1.6)


Kn-2 + Ln-2 = 0 (kn)


(n > 2)


Thus


W(Z) iV(Z) = (Mk2log k) + 2iK1Z


in the annulus (70,C).

Finally, according to equation (11.7)


A = -K1 + o(Mkl3)


Jn = n 1 Knl + O n+2
n


Q U(Z) = -R(Kl)log Z 2K2Z + n + 1 Kn+ +1
2 -- n Zn
1

Summing up: If one is satisfied with defining V(Z)
for O(Mk2log k) and U(Z) except for O(Mk3log k),
the corona (o',Co)


O(Mk3log k)


and W(Z) except
one may write in


W(Z) = iV(Z) + 2iKiZ

V(z) = H(Z) K1Z


(11.8)


Thus






NACA TM 1354 37


with


H(Z) = Kn (II.10)
1

and


U(Z) = Z dZ 2K2 (II.11)


The coefficient K1 may be supposed to be real, and the integra-
tion occurring in equation (II.11) must be made in such a manner that
R[U(Z)] will be an infinitely small quantity of the third order at
least on II = 1.

2.1.2.4 Remarks.

(1) The formula (1.8) which is the most important may be estab-
lished immediately from the second formula (1.25). However, the method
followed in the text, even though a little lengthy, seems to us more
S natural; also, it shows more clearly the developments of the func-
tions U, V, W.

(2) Strictly speaking, the hypotheses set forth in the course of
this study must be verified by the solutions found in each particular
case. We shall, however, omit this verification which in the usual
cases is automatically satisfactory.

(3) The results obtained by the preceding analysis and condensed
in the formulas (II.8), (II.9), (II.11) are in all strictness valid only
in the annulus (o0'CO), but not in the domain (D). However, it is
very easy to extend, by analytical continuation, the definition of H
to (D). Let us now first suppose that (C) contains 0 in its
interior; since one may write V(Z) in the form


V(Z) = H(Z) Z KnZ + B log Z
1

one sees that, since V(Z) is defined by hypothesis in (D), and one

can extend KnZn and B log Z inside of (70) up to (C), H(Z)
1






NACA TM 1354


may itself be defined without difficulty inside of (D). The case where
(C) does not contain the origin offers no difficulty; it is then suffi-
cient to utilize the development given at the end of section 2.1.2.1.

As to the order of the terms neglected when one writes the equal-
ity (IT.9) in the domain (D), they are found to be O(Mk2log k) in
(D) in the case where there exists inside of (C) a circle of the
radius Xk (X and 1/k may be considered as 0(1)). Besides, if
that is not the case, one may justify the validity of the results of
the formulas (11.8), (II.9), (11.10), (II.11) by making a conformal
representation of the domain (D) on an annulus; the radius of the
image circle of (Co) may be assumed equal to unity; the image circle
of (C) has a radius infinitely small of first order with respect to
k and the study may be carried out in the new plane of complex variable
thus introduced, without essential complication.


2.1.3 Reduction of the Problem to a Hilbert Problem

If one puts, according to the formula (II.8)

V = v + iv'

with v' denoting the imaginary part of V, one may write on (C) the
relation

w = -v'


Since one may, of course, with the accepted approximations, neglect u
compared to 1 in the second term of the formula (1.28), one sees that
this boundary condition (1.28) affects now only one single analytical
function, the function V(Z); this is a first fundamental consequence
of the preceding study. Formula (II.9) shows that this condition con-
sists in posing a linear relation between the real and the imaginary
part of H(Z) on the obstacle. Now according to equation (II.10) the
function H(Z) is a holomorphic function outside of (C), regular at
infinity; the problem stated which initially referred to an annular
area (D) is thus reduced to a Hilbert problem for the function H
defined in a simply connected region; exactly speaking, one has to solve
an exterior Hilbert problem. This is the second fundamental consequence
of the results of section 2.1.2.

Since we attempt to calculate V(Z) and W(Z) not further than
within O(Mk2log k), and U(Z) within 0(Mk5log k), the relation (1.28)





NACA TM 1354


which is written


R (v iw)2Z2d0 i dz(l p2


may be simplified and reduced to


S- dZ(v iw) 2P d
[ ] 2


On (C),
and therefore


KiZ is, according to equation (II.1), of the order of Mk2,


H = V = v + iv' = v iw


consequently, H satisfies, on (C), the Hilbert condition


R iH(Z) dZ] 2p- d6


(11.12)


2.1.4 Solution of the Hilbert Problem

A function H(Z), holomorphic outside (C), regular and zero at
infinity, satisfying on (C) the relation (II.12) must be-found. Let


a1
z = Z + a+ +
S Z


(11.13)


be the conformal canonical representation of the outside of (C) on
the outside of a circle (7) centered at the origin of the plane z;
the adjective canonical simply signifies that z and Z are equivalent
at infinity.

On (7) we shall put


z = reil


S2p2 d
. -de






NACA TM 1354


r being constant and well determined. Let us put


F(Z) = i log z (II.14)


One has on (C) or on (7)


F'(Z) dZ = i dz = -dP = f(e) dO (II.15)
z

with f being real; consequently



d d F'(Z) d F'(Z)
P-- -,z =~ 4i


and therefore equation (II.12) is written


H(Z) p2
S2 (11.16)
l F'(Z) p dI

H(Z)/F'(Z) is a holomorphic function outside of (C) and regular at
infinity. Following a classical procedure, we thus have reduced the
Hilbert problem to an exterior problem of Dirichlet.

Let G(Z) be the holomorphip function outside of (C), real at

infinity; its real part assumes on (C) the values G(Z) is
0 dQP
determined in a unique manner. According to equation (II.12)

H(Z) = -iG(Z)F'(Z) + iEF'(Z) (11.17)

with c being a real constant.

However, we have seen (section 2.1.2.3) that the coefficient of 1/Z
in the development of H(Z) around the point at infinity (coeffi-
cient KI) was real; now, around the point at infinity

a
iF(Z) =- + +.
Z 2





NACA IT 1354 41


In order to have the development of the second term of the formula (11.17)
admit a real coefficient of 1/Z, E must be zero since G(Z) is real at
infinity. Thus the desired solution is

H(Z) = -iG(Z)F'(Z) (II.18)

With the function H(Z) thus determined, the formulas (11.8),
(11.9), (II.11) permit calculation of the complex velocities U(Z),
V(Z), W(Z) within the scope of the accepted approximations. Thus the
problem posed in section 2.1.1 is solved.

Remarks.

(1) Uniqueness of the solution.- The preceding reasoning shows the
solution of the Hilbert problem satisfying the conditions (II.16) to be
unique. This result will be valid for our problem if one shows that
every function satisfying the condition (II.16) is a solution of the
initially posed problem (condition (II.14)) which is immediate since it
suffices to repeat the calculation.

(2) Calculation of the coefficient K1.- According to what has been
said above, the coefficient K1 is equal to the (real) value assumed by
SG(Z) at infinity. In order to find G(Z), we may solve the Dirichlet
problem in the plane z; according to a classic result of the study of
harmonic functions, K1 is equal to the mean value of 2p2 O on the
circle (7). Hence

K1- = 2f 2p d d9P = i-f p2d = 2S
27 0 d dp PJ() Tr

wherein S represents the area inside the contour (C).

2.2 Applications

2.2.1 General Remark

Let us consider a cone of the apex 0 in the space (Oxl,x2,x3),
the image of which in the plane Z is the curve (C), defined by its
polar equation p(e). According to the definition of p (see the
remark of section 1.2.5) the sections of this cone made by planes par-
allel to Ox2x3 are homothetic to the curve






NACA TM 1354


x2 2p cos e x 2p sin 8 (II.19)
1+ p2 1 + p

In the case of the linear approximations, with grad u, grad v,
grad w being infinitely small (it would even be sufficient that they
should be limited), one sees that one may, within the scope of the
approximations of section 2.1, simplify the formulas (11.19) without
inconvenience and write them


x2 = 2p cos 6 x3 = 2p sin 0


hence the result, essential for the applications.

The curve (C) in the plane Z is homothetic to the sections of
the cone obstacle made by planes normal to the nondisturbed velocity.

Let us likewise consider a cone with variable but small incidences
so that the flow about the cone should always be a flow in accordance
with the hypotheses of this chapter. One sees that if the orientation
of the cone varies with respect to the wind, the curve (C) in the
plane Z undergoes a translation.


2.2.2 Study of a Cone of Variable Incidence

This last remark allows us to foresee that when a thorough investi-
gation of a cone has been made for a certain orientation with respect to
the velocity it will not be necessary to repeat all the work for any
other orientation. This we shall specify after having demonstrated the
following lemma.

2.2.2.1 Lemma.- One may write on (C) that


2P2 de 2 R Z dZ (II.20)
0P d r;, ~d (11.20)
P dT p dz


Actually, let us put


Z = p cos 0 + ip sin e = X + iY


X and Y may be considered as functions of :P.


42






NACA TM 1354


Hence one deduces that


Y
tan e =
X


de
cos2


Y'cpX X'rpY
X2--
x2


and consequently


2 p2 de = 2 Y' X X'rY) = 2 R
P a;P 134 p/


- i dZ 2 R dZ
d;' R dz


which establishes the formula (11.20).

2.2.2.2.- Let us now consider two contours (CO) and (C1) defined
in the plane Z by two functions Z(O)(-') and Z(1)(c) such that
Z(O) = Z(1) + a, a being a complex constant determining the change in
orientation. In the development (II.13) which gives the conformal repre-
sentation, only the coefficient a0 varies when one passes from the
contour (CO) to the contour (Cl). Consequently


,z (1)
dz


dZ(o)
dz


and the Dirichlet condition determining the function GC"'(z) is written
in the plane z


RE()z) =2 R z(l) z] L )z +- Raz
dz_ 0- dz] dz

(we have omitted superscripts for the quantities which retain the same
value, affected by the index 0 or 1). Consequently


G(1)(z) = G(0)(z) + [g(z)


since g(z) is a regular function and real at infinity, holomorphic out-
side of (7), the real part of which on (7) assumes the






NACA TM 1354


values R(az dZ/dz), g(z) is then very easily determined. One has
exactly


g(z) = z dZ + ar2
\dz / z


Thence for the function H(1)(z), (since


F'(Z) = i/z dz/dZ)


H(1)(z) = H(0)(z) + a dz + 2
0 \ dZ( -2 dZ


(11.21)


The formula (11.21) gives immediately the solution of the problem of
change in orientation with respect to the nondisturbed flow.


2.2.3 Cone of Revolution

We shall study first of all the case of the cone of zero incidence.
One may then do without the preceding analysis and obtain the solution
directly; that is what we shall do here. The curve (C) is a circle of
the radius p = cte = r; the relation (1.28) is written


2rO
v cos e + w sin 8 = 2ro
P(1 + r02


On the other hand, for reasons of symmetry

v sin 0 w cos e = O


Hence one deduces immediately the values of v


2r0 cos 0
S=
b0 1 r02


and w on (C)


2r0 sin 0

1 r02)






NACA TM 1354


whence


v(z) = 2ro 2 Z
3(1 -)- r


W(Z) = i 2ro 2
0 1 ro


+ )


(11.22)


Finally the relations (1.25) permit the calculation of U


- dU 2r02 2Z 11 + Z2 _
P(l r) Z2 +1 Z2


4 rO2 1
S1 ro4 z


2
U(Z) 4 r0 log Z
2 1 r4


(11.23)


We shall now study, returning to the method of section 2.1, the
case of a cone of revolution with incidence.

The formula (II.13) is written

z = Z a

a being a constant which may be supposed to be real.

Consequently


F'(Z) = i


On the other hand, an immediate calculation shows that


do r(r + a cos T)
dP p2


whence






NACA TM 1354


and consequently


whence


2p2 dO 2 (2 + r cos rP)
P d'P (\



G(Z) = (r2 + ar2)
P\ Z a


According to equation (11.18)


H(Z) = 2 + ar2 ) 1 2 r2 Z
\ Z aZ a p (z a)2


the calculation is easily accomplished; one finds


V(z) 2r2 Z 1
(Z a)2


U(Z) = 4 og(Z a)
02


3_a a2
Z a (Z a)2


K2 = +a r2


In particular, one finds, if a = 0, by means of the approximate
formulas (11.24) and (11.25), the same result as by the formulas (II.22)
and (II.23) under the condition of neglecting in these formulas the term
in r0 of the denominator.

In order to give to these formulas a directly applicable form it
suffices to again connect the quantities a, r with the geometrical
data; for this purpose, one must use the formula defining p (p. 42).


since


+ 4aZ


(11.25)






NACA 3T 1354


Figure 11 represents the cone section made by the aerodynamic plane
of symmetry; a is the semiangle at the apex, 7 denotes the angle of
the cone axis with the nondisturbed velocity.

One has immediately

2r = pa 2a = py

Finally, we shall utilize for the calculation of Cp the for-
mula (I.11) since the velocity component u is infinitely small com-
pared to the components v and w. This formula is here written


Cp = -2R[U(Z) V(Z) 2 (11.26)


According to equations (II.24) and (11.25) one has


C = 2a21og ~2 2 + 4ay cos 0 + 272cos 20 (II.27)
jp Pa

The case of the cone of revolution of zero incidence is obtained
by making 7 = 0. One finds then again a known result. The for-
mula (II.27) had already been given by Busemann (see ref. 9) without
demonstration.


2.2.4 Elliptic Cone

We assume first of all the simplest hypotheses where the
planes Oxlx2, Oxlx3 are symmetry planes of the flow (U is in the
direction of the cone axis), with the cone flattened out on Oxlx2.
The formula (II.13) may be written in the form


Z = z + a2
z

or


p cos 6 + ip sin = r + a2 cos 9 + ir a)sin 9
r r \






48


Hence one deduces successively


tan B = r2 a2 tan P
r2 + a2


Scos2 r2 a2
cos2m r2 + a2


NACA TM 1354


r2 a 1
r2)2
r


2 2 dO = 2/2 a
0 d' P r2


The Dirichlet problem, which permits calculation of G(Z), is
readily formulated; since G(Z) has a constant real part on the con-
tour (C), G(Z) is constant:


z2


F'(Z) = i 1

z z 2


whence


H(z) = r2







H(Z) 2
(r


_ a2 a2
z





aA 1
r2Z2 4a2


We note besides that


and


K2 = 0.





NACA TM 1354


One calculates V(Z) by the formula (II.9)



(14a
V(Z) = 2(r2 1 Z)
\ r2/ ^" j2_


and U(Z)


by the formula (II.11) which may also be written


U(Z) = H K G 2K2Z
0 1 z '


whence


U(z) = i2 r
p2\(


U(Z) 2
p2


lo z z2 a2


Z + Z2 4a2
2


If one makes a = O, one will find again the expressions already
obtained for U(Z) and V(Z) in the case of a cone of revolution of
zero incidence (formulas (II.24) and (II.25) in which one makes a = 0).


We shall denote by e and by Tj
elliptic cone (see fig. 12). One has


the principal angles of the


E3 = 2(r + a2


T = 2(r a)


whence


r = (E + r1)
4


(11.28)


(11.29)


or


(11.30)


- l
r 2


a2 =
16


(C2 2)






NACA TM 1354


The pressure distribution on the cone circumference is easily cal-
culated. It is sufficient to apply the formula (II.26); besides


IV(Z)2 = E2112
T2cos2p + E2sin2cp


R[U(Z]- = log C+ T


2 2sin2(2
r2cos2C + E2sin2rg


hence the final formula


- 1 +


The case where the velocity is
may be treated equally by utilizing
one must put


HO(z) = (2 a 1--
-< -


One then obtains


H(l)(z) = 2(r2
(r


r z2 a2


not in the direction of the axis
the formula (11.21). In this formula



dZ= 1 a2
a2 dz z2


z2
z2 a2


zr2 z2
z2 z2 a2


2 kr2 E- z + r2 a2]
P(z2 a2) k 2

hence, remarking that


a2
Z = z + -- + a
z


and


(11.31)






NACA TM 1354


H(Z) = 2 2 1
\ r T(Z )2 4a2


(ax2 a2(z a(- (Z -a)2 4a2

2a2 (Z a)2 a2


V(Z) = H(Z) jr2 Z
-2]



On the other hand, we shall calculate U by utilizing the vari-
able z and the formula (II.20). The coefficient K2 is equal to



K2 ( r2 a + r2 a 2a


and U(z) is then given by the formula


a4 og 2a2 + az
_ log z z
r2/ z2 a2


+ a2 2 (22 + a)
z(z2 a2)


One will note that, if
mula (11.30), and that, for
except for the notations.


one puts a = 0, one finds again the for-
a = 0, one finds again the formula (11.25),


Thus one can, without any difficulty other than the lengthy writing
expenditure, calculate the pressure distribution coefficient on the
elliptic cone of any arbitrary orientation with respect to the wind.


U(z) =- 4 (r2
P2


K2
+4 -Z
p


(11.32)






52 NACA TM 1354


2.2.5 Calculation of the Total Forces

We have already seen in section 1.2.6 that the normal to the conical
obstacle directed toward the outside has as direction parameters


i(x3x2' x2x'), x3', -x2'


Let n be the unit vector coincidental with this normal, s be
the area of the section with the abscissa xl, L the length of this
section; one may make correspond to the resultant of the forces acting
on a section (x4, xl + dxl) a dimensionlesss) vector


Cz = C pn ds (11.33)


situated in the plane x2x3, and a dimensionless number



C 1 Cp(nU)ds (11.34)


the vector Cz characterizes the lift, the number C, the drag.

The integrals appearing in the formulas (33) and (34) are taken
along the section. Naturally Cz and Cx are independent of this
section. One may also replace C by a complex number Cz, the real
and iaginary parts of which are equal to the components of the vec-
tor C on Ox2 and Ox3. For calculating equations (11.33) and (1.34
one may utilize the section xl = 8. If we assume I to be the length
of the contour (C) in the plane Z, we may write, taking into account
the habitual approximations


Cz Cp dZ (11.35)


and

Cx [i CpZ dZ (I.36)
51 JC J







NACA TM 1354


with the integrals appearing in equations (11.35) and (11.36) taken in
the plane Z. These integrals present a certain analogy to the Blasius
integrals (ref. 13); Cp is given by the formula (II.26); unfortunately,
it is not possible to give simple formulas for the total forces since
the integrals (11.35) and (11.36) make use of all coefficients of the
conformal representation19.


We shall apply
circular cone; Cp


the formulas (11.35) and (11.36) to the case of the
is given by equation (11.27)


dZ = i fB eide
2


Z dZ = i 24- de
4


2 = xSp


One obtains


C, = -2ay


In the case of the elliptic cone of zero incidence, Cz
ously zero


- -i- a2 eiP
Z = re-i+P + e-
r




Cx = 2( r2
0P


Cx= 2a31og 2- 3 My2
OM


(11.37)


is obvi-


ireicP a2 iPj
dZ = i e e- doP
r




- 21 Cp dCP
Z^O p


being given by formula (II.31). Now


ScrT dP =
2(Tl2cos2% + e2sin 2 )


+ E dt
Do T2 + E2t2


19See appendix No. 7.


whence


with Cp






54 NACA TM 1354


As one can see immediately by putting

t = tan CP

the calculation of this last integral is immediate.

Thus one obtains


c = 2 2l g (E + n) 2 (11.38)



with 2 being the length of the ellipse with the semiaxes C-, -.
2 2


2.2.6 Approximate Formula for the Calculation of Cx

Let us consider the function U(z); according to formula (II.11)
and the remark 2 of section 2.1.4 one may say that the principal term
for U(z) is


U(z) = 4 Si log z
rp2


Consequently, in first approximation

8s
C S log r


with S being the area inside of the contour (C), and r the radius
of the circle (7) on which one makes the conformal canonical repre-
sentation of (C). If one now calculates Cx, taking into account this
approximate formula, one has, according to equation (11.36)


C 1S log rR i dZ


whence

C, =+ 32S2 log_ (II.39)
itp3" r






NACA TM 1354


We shall state: In every first approximation the value of the
drag coefficient C, is given by the formula (II.39).


2.2.7 Case Where the Cone PresenLs

an Exterior Generatrix

If the contour (C) shows an exterior angular point, the various
functions introduced in the course of the study (first paragraph of
this chapter) present certain singularities. These singularities we
shall specify. Let ZO be the designation angular point of (C), and
B6 the angle of the two semitangents to (C) at the point Z0(O < 6 < 1)
(see fig. 13); if- z0 is the image of the point Z0 in the plane z,
one may write, according to a well-known result, in the neighborhood
of zO


(LdZ\ K= z zo)k
\dz10


with K being a complex constant and k = 1 8; consequently

k
F'(Z)0 = K1z -0 = 2 (Z ZO 1+k


with K1 and K2 being complex constants. F'(Z) thus becomes infinite
at the point Z = Z0.

In contrast, the function G(z) has, according to definition, a
real part which assumes on the circle (7) the values


RzZ -]


This real part thus remains finite on the circle (7) (and it
satisfies there a condition of Holder). According to a known theorem,
Sits imaginary part likewise remains continuous on (7) (and likewise
Satisfies a condition of Holder). Consequently, one sees, if one refers
Sto formula (11.18) that






56 NACA TM 1354


k
H(Z) = K3(Z ZO) l+k


in the neighborhood of ZO; likewise, U, V, W will, in the proximity
of this point, be of the order k with respect to 1
1 + k Z- Z

Thus the analysis made in section 2.1 is no longer applicable to
this case. However, the formulas (11.35) and (II.36) show that if the
pressure coefficient assumes very high values in the neighborhood of
Z = ZO, the total energy remains finite. According to what we have
indicated in section 1.1.3 we consider the solution still valid, with
the understanding that the values of Cp in the surroundings of Z = ZO
are not reliable.


2.2.8 Delta (A) Wing of Small Apex Angle

at an Infinitely Small Incidence

If one puts in the formulas r2 = a2, at the end of section 2.2.4,
one obtains the pressure distribution on a delta wing with small apex
angle. Let us recall that a delta wing is an infinitely small angle.
Its angle, according to definition, is the half-angle w at the vertex
(compare fig. 14). Thus one has

op = 4a

The formulas (11.31) and (11.32) are applicable to a delta wing of
small angle placed at an incidence also rather small.

Let us moreover assume that this opening is infinitely small with
respect to the incidence. Under these conditions, the formulas yielding
U(Z) and V(Z) are written


V(Z) 2 a a Z 42 42
P2 2 4z2 4a2


2 2
U(Z) 4 a 4a + 8a (a a)Z (11.40)
82 2 JZ2 _4a2 02






NACA TM 1354


Actually one is justified in omitting the second-order terms with
respect to a. For calculating Cp it suffices to apply the for-
mula (11.8); the second term of the second formula (II.40) may be
neglected.


With the incidence 7, the delta wing being parallel to


70 = 2ia


Finally, one may put along the A


Z = 2a cos q = R cos (P
2


One then finds


c = 2'uy
C s-in
S sin


Ox2, one has


We remark further that p is related to the angle


213 = mP cos i


I of figure 14


W = n cos
2


One may state: the pressure coefficient on a delta wing of infi-
nitely small opening angle is independent of the Mach number of the flow.

One has


Cp = 2 if 2i
S-_ 2 tu
uI t


if one applies formula (11.35), one finds


Cz = inay


This coefficient Cz has not the same significance as the one
utilized in the theory of the lifting wing. Actually, it is, according


(11.41)






58 NACA TM 1354



to the very manner in which it was obtained, relative to the total area
of the A (pressure side and suction side); if one takes only one of
these areas into account, one must write (neglecting the factor -i)


Cz = 2nwu


This formula has been found by other methods by R. T. Jones
(ref. 14). We shall find it again in chapter III, section 5.1.2.4, when
studying the general problem of the delta wing which is here only touched
on incidentally and for the particular case of a A with infinitely
small opening angle.


2.2.9 Study of a Cone With Semicircular Section

As the last application, we shall treat the case of a cone with
.--
semicircular section, with the velocity U being directed along the
intersection of the symmetry plane and of the face plane of the cone20
(fig. 15).

The contour (C) in the plane Z then is a semicircle, centered
at the origin, of the radius a (fig. 16).

One obtains very easily the conformal canonical representation of
the exterior of this contour, on the outside of a circle (7) of the
radius r, centered at the origin of the plane z, by means of a par-
ticular Karman-Trefftz transformation (ref. 13, p. 128) which is written


-i 2
S6
Z a z re (11.42)
Z + a 5n
-1-i
z re

a and r are connected by the relationship


4a = 3r41


In order to obtain the correspondence between the circle (7) and
the contour (C), one must distinguish two cases. Let us put

z = re


20Such a cone formed the front of supersonic models planned by
German engineers.






NACA TM 1354


(1) < 6 < the corresponding point of
of the circle.

Let us put under these conditions


Z = aeir

and we shall find according to formula (II.42):


(C) is on the arc


tan -
2


(11.43)


(2) 7 < p < n, the
6 0
ment AA'; let us put under


corresponding point of
these conditions


(C) is on the seg-


Z = a cos .


The formula (11.42) shows that


r4
tan -
2 P 5t
tsin( + 12
sin(+ j


(II.44)


The two last formulas define completely the desired conformal
representation. Figures (17) and (18) give the variations of and x
as functions of P.

We shall have to utilize equally the value of dz/dZ. The simplest
method for obtaining this value consists in logarithmic differentiation
of the two terms of formula (II.42). One thus obtains the result


dz z2 + rz r2
dZ 2 2
Z -r


(I1.45)






NACA TM 1354


If one has


- < ( < 6 6


z = rei


Z = aei-


whence


dz r2 1 + 2 sin q ei('-A) 8 1 + 2 sin 'P ei((P-)
dZ 2a2 sin 27 sin *


71r 117
If 'p is comprised between -- and --, one puts
b b
Z = a cos X. Thus one obtains


i(E --A
dz 16 1 + 2 sin r e2 )
dZ 27 sin2X


The function G(Z)


has as its real


part R zZ ]d that is
[- d~dz


2 ar sin 4
8 1 + 2 sin T


0


if <'JP <
6 6


if 6 < < 1
b o


The analytic function


a2 z dZ
Z dz


has a real part which, on (b), assumes these same values. This func-
tion is regular at infinity, holomorphic outside of (7), but with a
pole z = -i, with the corresponding residue being equal to -ia2
pole z = -ir, with the corresponding residue being equal to -ia


(11.46)


z = re ,


(1.47)


(II.48)






NACA HM 1354 61


Let us then consider the function


a2/2 dZ 1 z ir\
\Z dz 2 z + ir)


This function is holomorphic outside of (7). It is regular at
infinity; its value at infinity is equal to a2/2. On (7), these real
and imaginary parts satisfy Holder conditions. This function is there-
fore identical with the desired function G(z).

Hence one deduces according to equation (II.18)


and ac


t:


H(Z) ( r\a2 dz
Z dz 2 z + ir/ z dZ


cording to equation (II.19)


V(Z) = a2l Z 1
\Z 2 2z


Finally, the calculation of U(Z)
of formula (II.29)



G dz = a2log Z a z ir dz
2 jz + ir z


and


ZH a2/ 1 Z dz z ir
ZH2 z dZ z +
2 z dZ z + ir


Sa2 a2 z ir dz
Z 2z z + ir dZ


z ir dz\
z + ir dZj


may be carried out with the aid


= a2 (log Z + .1 log z)
\ z + ir 2 /





S= a2 z ir Z dz\
2) 2 z + ir z dZ/


whence


U(Z) = a2 ir Z dz 1 + + 2 log Z + log
\z + ir z dZ g2 z + ir


The calculation of the coefficients K2 offers no difficulty
soever; however, as one had already opportunity to note, the term
does not occur in the calculation of the pressures along the cone.


n


i;
ii
i;
iii
,i
II
tl;


,ii

,,
,;


what-

K2






62 NACA TM 1354


This pressure distribution along the cone calculated with the aid
of equation (II.26) is represented in figure 19.


2.3 Numerical Calculation of Conical Flows With

Infinitesimal Cone Angles

2.3.1 General Remarks

In the preceding paragraph, we have studied a certain number of
particularly simple cases. However, if the cone (C) is arbitrary, it
will be necessary to carry out various operations leading to the solu-
tion by purely numerical procedures.

Let us analyze the various operations necessary for the calculation:

(1) The conformal canonical representation of the exterior of (C)
on the outside of the circle (?) must be made; this calculation per-
mits, in particular, determination of the radius r of (7), corre-
spondence of the points of (C) and of (7), and calculation of the
expression dZ on the contour (7).

(2) The function G(z), holomorphic outside of (7), regular and
real at infinity must be determined, the real part on (7) of which is
known; we shall designate it by g('). In fact, it suffices to know,
on (7), only the imaginary part of G(z), for instance g'('P); g'(P)
is the conjugate function of g(T) and is given by the formula


g'() =_ 1 g( ')cot P- dc'
2n Jo 2

This formula is called "Poisson's integral."

(3) With these two operations accomplished, the values of H(z) on
the circle (7) (formula (11.18)) are known which provides the values
of v and w on the cone; u is obtained by the formula (11.29). The
only new calculation to be made is that of the expression:


[B- f dz I drP


the constant of integration being determined so that u should have a
mean value zero on (M).






MACA 9M 1354


All these operations always amount to the following numerical
problems:

(a) With a function given, to calculate its conjugate function
(Poisson integral)

(b) With a function prescribed, to calculate the derivative of the
conjugated function

(c) With a function prescribed, to calculate its derivative21.

We shall justify this result in the following paragraph by showing
that the operation (1) may be performed by applying the calculations (a),
(b), (c). We shall then indicate a general method, relatively simple
and accurate, which permits solution of these problems. We shall ter-
minate this chapter by giving an application.


2.3.2 Conformal Canonical Representation

of a Contour (C) on a Circle (y)

The numerical problem of determination of the conformal canonical
representation of a contour (C) on a circle (y) has been solved for
the first time by Theodorsen22. We shall briefly summarize the principle
of this method, simplifying, however, the initial expos of that author.

Let us suppose, first of all, that the contour (C) is neighboring
on a circle of the radius a, centered at the origin (fig. O0); in a
more accurate manner, putting on (C)


Z = ae+ie (11.49)

with being a function of e, 0 = 4(e), we shall suppose that *(e)
and d- are functions which assume small values. We shall then call
de

21If the conformal representation of the exterior of (C) on the
outside of (7) is known in explicit form, it will naturally be suffi-
cient to apply operation (a).
22Compare references 15 and 16. One may achieve this conformal
representation also by the elegant method of electrical analogies (ref. 17);
the time expenditure required by the experimental method and by the purely
numerical methods here described as well as the accuracy of these pro-
cedures are of the same order of magnitude.





NACA TM 1354


(C) "quasicircular." Let Q be the angular abscissa of the point of
(7) which corresponds to the point of (C), the polar angle of which
is 0; we put


e = P + c((p)


cP = 0 E(0)


e(e) and E(cp) representing the same function but expressed as a
function of 0 or as a function of qP; we shall put likewise

(CP) = J1(e)

The desired conformal transformation may be written

Z zeh(z)

with h(z) being a holomorphic function outside of (7), regular and
zero at infinity. The equality (II.50) becomes, if one writes it on the
circle (7),

ae())+i T +(' = reiqeh(z)

whence


h(z) = W(~() + iE(cp) + log
r r


(11.51)


Finding the conformal representation of (C) on (T) amounts to cal-
culating the functions (cp) and T(p). First of all, one knows (equa-
tion of (C)) that


(11.52)


(p) = + C(r4) )


On the other hand, according to equation (II.51), e((P)
gate function of (cp), and consequently


is the conju-


-)=(<'P)cot(' 2 (
N(c') ot M P'
k 2


(11.50)


e(') 1 '
2x p n
2v


(II.53)






NACA TM 1354


the integral being taken at its principal value. There is no constant
to add to the second term of equation (11.53), for i(P) has a mean
value zero since h(z) is zero at infinity. For the same reason, if
*0 denotes the mean value of T('o) in an interval of the amplitude 21


r = ae'0


(11.54)


an equality which will permit calculation of r if T(P) is known.
In order to calculate T(P) and T(P), one disposes therefore of the
relations (11.52) and (11.53); one can solve this system by a procedure
of successive approximations.


We shall put first

o(e) = T0() = o


According to equation (11.52)

4(e) = on(0)

and according to equation (11.53)


l(e) = 1


'(S')cot d' e de'
2


Thence a first approximation for 7'


1 = a l(e)


S=' 1 + 1 )


From it one deduces, according to equation (II.52), a first approxima-
tion for Tl(c')


11 1 +


whence a second approximation for the function E





NACA TM 1354


r2 1 02n 1 1cot 1' 2p1
1 = j*2y W (c1)cot %i l ,P
2a 2 2 T


2(e) = c2 [e e1()]

whence

2 = a E(e) e = 2 + 2n


The procedure can be followed indefinitely.

The convergence of the successive approximations forms the subject
of a memorandum by S. E. Warschawski (ref. 18). We refer the reader
who wants to go more deeply into that question to this meritorious
report.

From the practical point of view one may say that the convergence
is very rapid; two approximations suffice very amply in the majority of
cases; the different changes in variables which encumber the preceding
expose are very easily made by graphic method. Thus one sees that the
numerical work essentially consists in calculating twice the inte-
gral (II.53). This calculation is precisely the object of the prob-
lem (a) stated at the end of section 2.3.1.

If the contour (C) is not "quasicircular," one may make, first
of all, a conformal representation which transforms it into the "quasi-
circular" contour (C'); one will then apply the preceding analysis to
the contour (C'). For certain cases it will be quicker to use a direct
method. Let us assume, for instance, that (C) is a contour flattened
on the axis of the X (compare fig. 21) and for simplification that
X'OX is permissible as the axis of symmetry.

Let us suppose that X varies along (C) from -a to +a while
IY| remains bounded by ma (with m being, for instance, of the order
of 1/10); it will then be indicated to operate as follows:

We put along (y).



Z = [f(rP) + ig(






SNACA TM 1354

One has




or also


X('C) = f cos CP g sin r

Y(P) = f sin p + g cos r


f = X cos r + Y sin m

g = Y cos '" X sin cP


f(P) is an even function of r,, g(C') is an odd function


f(0) = +f(n) = a


g(O) = g(n) = 0


The functions X(T) and Y(0) have to be found. Let us take as
starting point

XO(0) = a cos P


an approximation which would be definitive if (C) were an ellipse.

On the contour (C) one reads the corresponding value YO('P), and
by means of the second formula (II.56) one obtains a first approximation

gl(+) = YO(cp)cos rp Xo(p)sin (p


fl(q) will be given by a Poisson integral


2xi
0


gl(r)cot -'- "- dT' + X1
2


with X1 being a constant, such as fl(O) = a.

Owing to the formulas (II.55), one has a first approximation Xl('),
Y1(T) for the functions X(), Y('P). One proceeds in the same manner,
reading off on (C) the functions Y1(mP) corresponding to Xl('), then


(11.55)




(I1.56)





NACA I 1354


calculating


g2 1) = Yl(c)cos (p Xl(p)sin C


and


n')
=f


g2( p')cot P' dTP' + X2
2


When one
approximation
tions; then


has obtained a pair fn(P), gn(') providing a sufficient
Xn(), Yn(p) of X(C'), Y('), one stops the calcula-


r = X


In practice23 it suffices to take
of very slight adaptations) will apply
being flattened on OX, will no longer
axis.


n = 2; the same method (averaging
to the case where (C), although
admit of OX as the symmetry


Finally, for a complete solution of the problem (1) posed at the
beginning of the preceding paragraph, only dZ/dz remains to be calcu-
lated, which will obviously be possible with the aid of the problems (b)
or (c).


2.3.3 Calculation of the Trigonometric Operators24

The method we shall summarize permits calculation of the linear
operators A, transforming a function P(e) into a function Q(e)

23The principle of this method is the one we applied for the study
of profiles in an incompressible fluid. But in the case of the profiles
a few complications (which can, however, easily be eliminated) arise due
to the fact of the "tip."
24We gave the principle of this method for the first time in
March 1945 (ref. 19). Compare also reference 20. In continuation of
this report, M. Watson provided a demonstration of the formulas which
we obtained by a different method (ref. 21). We also point out a "War-
time Report" of Irven Naiman, of September 1945, proposing this same
method of calculation for the Poisson integral (ref. 22).


etc.






NACA TM 1354 69


Q(e) = A[P(e)

and re-entering one or the other of the following categories:

First category: The operator possesses the following properties



A(cos me) = am sin me

A(sin me) = -a cos me (II.57)

A(1) = 0



with am being a nonzero constant, m any arbitrary integral different
from zero.

Second category: A possesses the properties



A(cos me) = bm cos me

SA(sin me) = bm sin me

A(1) = bO



with bm being a nonzero constant, m any arbitrary integral.

We shall call these operators trigonometricc operators." The
operations which form the subject of the problems (a), (b), (c) are,
precisely, particular cases of trigonometricc operators."

With the function P(e) known, one now has to calculate the func-
tion Q(e); the functions P(e) and Q(e) are assumed as periodic, of
the period 2v. P(e) and Q(8) are determined approximately by knowl-
edge of their values for 2n particular values of 8, uniformly dis-
tributed in the interval 0, 2n. One knows that the unknown 2n values
of Q are linear functions of the known 2n values of P. The entire
problem consists in calculating the coefficients of these linear equa-
tions. We shall do this, examining two possible modes of calculation.





NACA IM 1354


2.3.3.1 First mode of calculation.- After having divided the
circle into 2n equal parts, we shall put


f= f


(1) Operators of the first category.- Obvious considerations of
parity show that the Qi are expressed as functions of the Pj by
equations of the form


n-1
i = KpPip i-p)
1


(11.58)


We shall apply the
equations (11.58)


relations (II.57), that is, carry into the


P(8) = cos me


P(O) = sin me


We thus obtain 4n
equation


Q(e) = am sin me


Q(e) = -am cos me


equations which are all reduced to the unique


n-l
SKp sin p m -a
1


(I1.59)


This reduction is the explanation for the success of the method.
We have to determine (n 1) unknown Kp. For this purpose, we shall
write the equation (II.59), for the values of p varying from 1 to
n 1. The system remains to be solved. If one multiplies the first
2!, scn b2 i)th
equation by sin -E, the second by sin L, the (n 1 by
n n
sin(n 1)n, and if one adds term by term, one obtains a linear rela-
tion between the Kp, with the following coefficients of Kp






NACA IM 1354 71


n-1 n-1
Wpn 1 (-nIP Jfn (p + [Jn
sin m sin m -- Co cos m
n n 2 5 n 1 nn
m=l m=l


1 Sn P Cn[ P( + )_n'
= 2 E D n In]J

with

n-1 ( sin x
Cn(x) = cos mx = cos x
2 x
m=O sin


Thus the coefficient of Kp is zero if p / n, and equal to if
2
p= U-

Thence the desired value of Kp


n-l
Kp = a sin m (pI.60)
m=l

Let us apply this result to the calculation of the Poisson integral.

This integral defines an operator Q = A(P) of the first category
for which am = -1.

Consequently, the formula (II.60) is written

n-1
Kp = : sin m = Sn
mn n n (
1

if one puts

n-1 sin nx
n- (n 1)x s 2
Sn(x) = V sin mx = sin -2 l)x
1 sin 2
1 2





NACA TM 1354


Thus

K = 0 if p even

(II1.61)
1 cot pA if p odd
KP n 2n


(2) Operators of the second category.- The considerations of parity
permit one to write the general formula

n-1
Qi = KOPi + (Pi+p + Pi-p) + Kni+n (11.62)
1

Using the same reasoning as before, one is led to determine the coeffi-
cients Kp by the system


n-1
KO + > 2Kp cos m E + ( 1)% = b (11.63)
p=l

with m assuming the values 0, 1, 2, n.

Multiplying the first value by 1/2, the second by cos pu/n, the
third by cos 2?, the nth by cos (n 1)p! and the last by ( )2
n n
and adding them, one obtains a linear relation between the Kp, with the
coefficient of Kp being (p / 0, p / n)



2 + )P+LL n ] + Cn(P- ) 2


that is, n if u = p, and 0 if p.

The coefficient of K0 is



2 2 2 + Cn n






NACA TM 1354


The preceding conclusions remain valid, it is zero for i J 0 and equal
to n if P = 0; the same result is valid for Kn. Finally, one
obtains the general formula of solution


(I..64)


Let us consider, for instance, the operator transforming the func-
tion- P(G) into the function dQ/de, with Q being the conjugate func-
tion of P; it is an operator of the second category for which

bm = -m


Applying formula (11.64), one obtains

KO m
2


n- n
_1


If one notes that


1n


n-l
(x) = m cos mx
0


12 sin n
2 sin2 x
2


one sees that


Kp = 0


1
n cos
n


if p even


if p odd


- 1)P


p / 0


- x sin2 x
2)






74 NACA TM 1354


2.3.3.2 Second mode of calculation.- Examination of an important
particular case will show us that in certain cases it will be advantageous
to consider a second mode of calculation.

The method consists in replacing the function P(e) by a function
of the form

n
D(e) = an cos nO + bn sin nO (11.66)
0

for which the method is applied with the strictest exactness; the con-
stants an and .bn are such that Pi = $i. One operator of the first
category, one of the most important ones, is the operator of derivation
which makes the function dP/de correspond to the function P(O). If
we apply the first type of calculation, we shall replace ( -) by
\de/i

d(I ; now, it is precisely at the points 8 = in that the deriva-
(dei n
tives and show the greatest deviation. In contrast, we shall
de de
obtain a good approximation of the desired function by replacing


dP (2i + 1)n b d' (2i + 1)t
dO 2n dO 2n

We are thus led to the following mode of calculation: the circle
is divided into 4n equal parts; we shall put


fi = f(il)
\2n/

and we shall express the 2n values Q2i as a function of the 2n value

2j+l"

We shall limit ourselves to the operators of the first category.
The formula expressing the Q2i as a function of P2j+l is written

n
>2i =_ K(P2i+2p-l 2i-2p+l)
p=l






N ACA TM 1354


and we obtain for determination of the Kp the system



Ssin (2p l)m am
l 2n 2
p=l


with m varying from 1 to n.


Multiplying the first equation by sin(2P 1)--, the second by
2n


(20 l)2n th
sin( 1)2, .,the (n 1)th
2n


by sin (2P l)(n 1)n
by sin the
2n


(- 1
last by and adding them, one obtains a linear relation in
2
which the coefficient of Kp is


n-1
Z sin(2p -
m=l


1)mL sin(2p -
2n


n-1
>- cos(p u)L cos(p
1


+.- I )B + + C )P-P
"J 2


S ipn n-p + [L 1) + ( 1)


The coefficient is zero if p 6 I, and equal to E if
2


sin (2p l)mn
sin
2n


( 1)-1
2


p = P. Hence


(11.67)


This procedure may be applied to the calculation of the derivative
of a periodic function. In this case, am = -m. Applying formula (II.67),
one obtains


)n ( 1)P+P
2n 2






NACA IM 1354


Kp= ( 1)P-1 1 (1.68)
[ (2p l)xl
2n 1 cos ( n)
11 2n

2.-.3.4 Remarks on the Employment of the Suggested Methods.- In
order to convey some idea of the accuracy of the proposed methods we
shall give first of all a few examples where the desired results are
theoretically known.

Let us take as the pair of functions P(O), Q(e), the functions


p(e) = 4 cos 20 4 cos 9 + 1 Q(e) -4 sin 0(2 cos 0 1)
(5 4 cos 8)2 (5 4 cos 8)2

which are the real and imaginary parts, respectively, on the circle of
radius 1 of the function


f(z) 1 (z = ei)
(2z 1)2

One will find in figure 22 the graphic representation of the func-
tions P(8), Q(e) and of the derivative Q'(0) of this function, and
also the values of these functions for e = p- (with p ranging
12
between 0 and 12). Furthermore, one will find in figure 23 the values
of Q(6), calculated from P(e) as starting point, by the method just
explained (coefficients Kp, defined by equation (II.61)), and in fig-
ure 24 on one hand the values of Q'(8), calculated from P(e) as
starting point (from coefficients Kp defined by equation (II.65)),
and, on the other, these same values calculated from Q(S) as starting
point (coefficients Kp defined by equation (11.68)). One will see
that the accuracy obtained is excellent although the selected functions
show rather rapid variations. Such calculations by means of customary
calculation methods are a delicate matter; this is particularly obvious
in the case of the Poisson integral which is an integral "of principal
value." Systematic comparisons of the method of trigonometric operators
with those used so far have been made by M. Thwaites (ref. 23); they
have shown that this method gives, in certain calculations, an accuracy
largely superior to any attained before.

The calculation procedure, with the aid of tables like the one
represented (fig. 25) is very easy. One sees that one fills out this






NACA IM 1354 77


table parallel to the main diagonal of the table. With such a table,
about one and a half hours suffice for a Poisson integral if one has a
calculating machine at his disposal.

We have had occasion to point out that the accuracy of the method
obviously increases to the same degree as the functions one operates
with are "regular" and present "rather slight" variations. This leads
in practice to two remarks which are based on the "difference method"
and reasonably improve the result in certain cases. We shall, for
instance, discuss the case of the Poisson integral.

(1) If the function P(G) presents singularities (for instance
discontinuities of the derivative for certain values of 0), it will be
of interest to seek a function P1(O), presenting the same singularities
as the function P(e), for which one knows explicitly the conjugate
function Ql(6). One will make the calculation by means of the func-
tion P(O) P1(0); this function no longer presents a singularity.

(2) If the function P(O) has a very extended range of variations,
one will seek a function Pl(8) for which one knows explicitly the
function Q%(8) so that the difference P(G) P1(0) remains of small
value, and one will operate with this difference.

Finally we note that, if the calculation of the derivative of a
function P(e) as described above necessitates that P(O) be periodic,
one can always return to this case, applying, precisely, the "difference
method."


2.3.4 Example: Numerical Calculation of a

Flow about a Semicircular Cone

As an application, we have taken up again the case of the semicir-
cular cone studied in section 2.2.9. The function g(1) is given by
the formula (II.48), and g'(Q) will be calculated by a Poisson inte-
gral. Figure 26 shows the value g'(c) thus calculated compared to the
theoretical value.

25We wanted to test the accuracy of the proposed method by assuming
an extremely unfavorable case, without taking into account the singu-
larities presented by the function g(T). For a numerical operation of
great exactness, this particular case would have required application
of the lemma of Schwartz, with the contour (C) completed symmetrically
with respect to OX.






78 NACA TM 1354


It is then possible to calculate the representation of the pres-
sures, by calculating successively the function H, ZH, and the inte-
gral g' (p).

One will find the pressure distribution thus calculated in fig-
ure 19; one may then compare the result obtained by the calculation
method (for a very unfavorable case) with the result obtained
theoretically.






NACA TM 1354


CHAPTER III CONICAL FLOWS INFINITELY FLATTENED

IN ONE DIRECTION


The purpose of this chapter will be the study of conical flows of
the second type (see chapter I, section 1.2.6). Before starting this
study proper, we shall make a few remarks concerning the boundary con-
ditions. The conical obstacle is flattened in the direction Oxlx2.
Under these conditions, reassuming the formula (1.27)


x2 vx3' = (xx2' x2x3') (1 + u) (1.27)


one may say that it reduces itself, in first approximation, to


w2' = x2x3') (III.)


since x3, x ', v, u are infinitesimals of first order, while x2
and x2' are not infinitesimals. Under these conditions, one may say
that one knows the function w on the contour (C). On the other hand,
one may write, within the scope of the approximations made, this boundary
condition on the surface (d) of the plane Oxlx2, projection of the
cone obstacle on the plane. Let us designate, provisionally, the
value w by w(1)( 1x2x3) if one operates as follows


(1)
w(l) J1x=t),(tx(t = w(l)E1,x2Ct)J] + xj(t) xx,2(t,


With the derivatives of w being, by hypothesis, supposed to be of
first order, and the boundary equation written with neglect of the terms
of second order, the intended simplification is justified.

Various cases may arise, according to whether the cone obstacle is
entirely comprised inside the Mach cone (fig. 27), whether it entirely
bisects the Mach cone (fig. 28), whether the entire obstacle is com-
pletely outside the Mach cone (fig. 29), or whether it is partly inside
and partly outside the Mach cone (fig. 30). In each of these cases
there are two elementary problems, the solution of which is particularly







NACA TM 1354


interesting: the first, where the relation (III.1) is reduced to

w = constant = w0


which we shall call the elementary lifting problem (the corresponding
flow is the flow about a delta wing placed at a certain incidence); the
second, where the relation (III.1) is reduced to


w = wO for x3 = +0

w = -wO for x3 = -0


which we shall call the elementary symmetrical problem. This is the
case of, for instance, the flow about a body consisting essentially of
two delta wings, symmetrical with respect to Oxlx2 and forming an
infinitely small angle with this plane. It is also the case that will
be obtained, the section of which, produced by a plane parallel to Ox2xy,
would be an infinitely flattened rhombus. The fact that one obtains the
same mathematical formulation for two different cases indicates the
relative character of the results which will be obtained. In the case
of the symmetrical problem one may naturally assume that w is zero on
the plane' Oxlx2 at every point situated outside of (d).

Let us finally point out that very frequently the obtained results
do not satisfy the conditions of linearized flows; in particular, the
velocity components and their derivatives will frequently be infinite
along the semi-infinite lines bounding the area (d). However, we admit
once more that the results obtained provide a first approximation of the
problem posed above, in accordance with the remarks made in section 1.1.3
of chapter I.


3.1 Cone Obstacle Entirely Inside the Mach Cone

3.1.1 Study of the Elementary Problems

The case of the lifting cone has already formed the subject of a
memorandum by Stewart (ref. 10); however, the demonstration we are going
to give is more elementary and will permit us to treat simultaneously
the lifting and the symmetrical case.






NACA TM 1354 81


3.1.1.1 Definition of the function F(Z).- We shall make our
study in the plane Z. Let A'A(-a,+a) be the image of the cut of the
surface (d)26, (Co), as usual, the circle of radius 1 (fig. 31).

Naturally, we shall operate with the function W(Z). One of the
conditions to be realized which we shall find again everywhere below is
that dW/dZ must be divisible by (z2 1), unless the compatibility
relations show that U(Z) would admit the points Z = 1 as singular
points which is inadmissible. Thus we introduce the function


F(Z) Z2 dW (III.2)
Z2 1 dZ

iand we shall attempt to determine F(Z) for the symmetrical as well as
for the lifting problem.

F(Z) is a holomorphic function inside of the domain (D), bounded
by the cut and the circle (CO); the only singular points this function
can present on the boundary of (D), are A and A'; on the other hand,
'iF(Z) must be divisible by Z2, unless U, V, W have singularities
at the origin. On the two edges of the cut F(Z) must have a real zero
part. On the circle (Co)


Z 1 1
Z2 1 Z 1 2i sin 0
Z

Z dW = e1 dW dW
dZ dZ dO


Consequently, F(Z) has a real zero part on (Co) as well. The fact
that F(Z) cannot be identically zero, and that its real part is zero
on the boundary of (D), admits A and A' as singular points. We
shall study the nature of these singularities.

3.1.1.2 Singularities of F(Z).- Physically, it is clear that A
Sand A' cannot be essential singular points. Let us therefore suppose
.that, in the neighborhood of Z = a, one has


26One assumes, as a start, that the problem permits the use of the
plane Ox1x3 as the plane of symmetry.






NACA TM 1354


F(Z) ~ K%(Z a)mo


m0 being arbitrary, KM / 0; let us put


Z a = rei


with Pq being equal to +- on the upper edge of the cut, to
the lower edge; for sufficiently small values of r


-it on


orO r" O"


and


KmO e-imo
W)e


must be purely imaginary quantities; thus the same will hold true for


KO cos molt


and for


iKm sin mon;


K 02 = Km02cos2m~ (i0 sin ms)2


is therefore real. On the other hand


2 sin 2mov
2 i2mus = (b cos n)( w ei sin n0g)

must be real which entails


sin 2m0A = 0


Thus there are two possibilities; let us denote by
integral; either


k an arbitrary


KmO is purely imaginary


or else


KmO is real.


mo = k + -,


mo = k,






SNACA TM 1354 83


Let us now consider


F1(Z) = F(Z) K(Z a)m0


In the neighborhood of Z = a


Fl(Z) ~ Kml(Z a)ml


and the same argument shows that 2mI must be an integral. Finally,
one may state the-following theorem:

Theorem: Inside of (Co) the function F(Z) may assume the form


F(Z) = O(Z) + 1 t(Z) (III.3)
a2 Z2

'with Q(Z) and 4(Z) admitting no singularities other than the poles

at A and A'.

The analysis we shall make will be simplified owing to certain
symmetry conditions which F(Z) satisfies. Let us put

W = w + iw'

Obviously, X in w(X,Y) is even (when Y is constant).

Consequently, F(Z) has a real part zero on OY. Applying
Schwartz' principle one may write


F(Z) = -F(-Z) (1I.4)


This equation shows that knowledge of the development of F(Z)
around Z = a immediately entails knowledge of F(Z) around Z = -a.






NACA TM 1354


3.1.1.3 Study of the case where F(Z) is uniform [c(Z) = -O.-
Let us consider the function


iz2P
Ap(Z) = iZ2
(a2 z2)(1 a2z2


with p an integral and >1.
This function satisfies all conditions imposed on F(Z).
Indeed, it satisfies equation (III.4); inside of (Co) it does
not admit singularities other than a and -a which are poles of the
order pl. Its real part is zero on the cut as well as on (CO), as
one can see when writing


Ap(Z)


2 (Z2 + )
a ( +
\ Z2~i


- (1 + a


Finally, the origin should be double zero (at least).
Let us assume F(Z) to be the general solution of the problem
stated; we shall then demonstrate the following theorem:
Theorem: If F(Z) is uniform, one has


n n
F(Z) = XpAp(Z) = i
11 ^a2


(III.6)


- Z2)1 a2Z21


with n being an integral, and the kp being real coefficients.

In case F(Z) is assumed to be a solution of the problem having a
pole of the order n, one can determine a number kn so that

Fl(Z) = F(Z) nAn(Z)

will be a solution admitting the pole Z = a only of an order not
higher than (n 1) at most. But in consequence of equation (III.4),


(III.5)






NACA TM 1354


F1(Z) will allow of Z = -a as pole of, at most, the order (n 1).
Proceeding by recurrence, one finally defines a function

n
Fn(Z) = F(Z) XpAp(Z)
1

which must satisfy all conditions of the problem and be holomorphic
inside of (CO). The boundary conditions on the circle and on the cut
entail Fn(Z) to be a constant which must be zero because Fn(Z) must
become zero at the origin.

3.1.1.4 Case where O(Z) = 0.- We shall study the case where
O(Z) = 0 in a perfectly analogous manner.

Let us put


f(Z) = F(a2 z)(l a2Z2) F(Z)
Z

f(Z) is a uniform function inside of (Co) which admits as poles only
the points (Z = -a, Z = a). Actually, the origin is not a pole since,
according to hypothesis, F(Z) is divisible by Z2. The function f(Z)
possesses the following properties: It is imaginary on the cut, real
on (Co), and real on OY (which entails properties of symmetry if one
changes Z to -Z). Moreover, f(Z) admits the origin as zero of, at
least, the order 1. All these properties appertain equally'to the
functions



$r =(z 1 iZP-1 2 1)
Bp(Z) = Ap 1) = Z-(Z2 ) p
Z Z) a2 2)(1 a2Z2

p is an integral >1.

Thus one deduces, as before, the theorem:


Theorem: In the case where O(Z)


= 0, one may write






NACA TM 1354


F(Z) = i n Z2pP (III.7)
z21z 2
1 a2 Z2)(1 az2] 2


with n being an integral, the Ap being real.

3.1.1.5 The principle of "minimum singularities".- The for-
mulas (III.6) and (III.7) depend on an arbitrary number of coefficients.
The only datum we know is the wo, the value w assumes on the upper
edge of the cut. Thus we have to introduce a principle which will
guarantee the uniqueness of the solution of the problems we have set
ourselves. This principle which we shall call principle of the "minimum
singularities" may be formulated in the following manner (it is con-
stantly being employed in mathematical physics):

When the conditions of a problem require the introduction of func-
tions presenting singularities, one will, in a case of indeterminite-
ness, be satisfied with introducing the singularities of the lowest
possible order permitting a complete solution of the posed problem.

In the case which is of interest to us, this amounts to assuming
n = 1 in the formulas (III.6) and (III.7). For the problem of interest
to us, this principle has immediate significance; it amounts to stating
that F(Z) and hence dW/dZ must be of an order lower than 2 in
1/Z a, or W(Z) must be of an order lower than 1 with respect to that
same infinity; the considerations set forth in section 2.2.7 show that
these conditions entail the total energy to remain finite.

3.1.1.6 Solution of the elementary symmetrical problem.- Let us
turn again to formula (III.6); one deduces from it, according to equa-
tion (III.2), that in the case where F(Z) is uniform


dW Z2 1
ik
dZ 41 (a2 Z2)(i a2Z2)


and hence


W(Z) = 1 log (a Z)(1 aZ)
2a(l + a2) (a + Z)(1 + aZ)






NACA TM 1354


The determination of the logarithm is just that the real part of W(Z)
is zero on (Co). Besides


2a(l + a2)wo


On the upper edge of the cut

w = w0


and on the lower edge w assumes the opposite value. This shows us that
the case investigated is that of the symmetrical problem. The value W(Z)
for this problem is therefore


W(z) IW- lo(a Z)(1 aZ)
i (a + Z)(l + aZ)


(III.8)


The calculation of the functions U(Z) and V(Z) offers no diffi-
culty whatsoever. It suffices to apply the relationships of compati-
bility (1.25) and to integrate; the only precaution to be taken consists
in choosing the constant of integration in such a manner that the real
parts of U and V on (CO) become zero; one then finds


v(z) -~oW (1 + a2)






u(z) -2
p(1 a2


(a + Z)(1 aZ)
g(Z a)(l + aZ)




Z -a2
S log a! -
) 1 a2Z2]


This last formula is the most interesting one since it permits calcula-
tion of the pressure coefficient (see formula (1.8)). One finds


4C W a [loga2-X2]
p p 1 a2 1 a2x2


(III.9)






(III.10)


(III.11)







88 NACA TM 1354


In order to interpret this formula, one must connect the quanti-
ties a, X, to geometrical quantities, related to the given cone. Fire:
of all

w0 =a


a being the constant
hand


inclination of the cone on



2X = r -= tan u
1 + X2 x


Ox. On the other


whence


= cos w 1 M2sin2
0 sin w


(see fig. 32) and


cos CU \ M2sin2w0
p sin a0


In figure 33 one will find the curves giving the values of
functions of o, for various Mach numbers and various values of


(III.12)


Cp as
CO'


3.1.1.7 Solution of the elementary lifting problem.- If one
starts from the formula (III.7), one obtains


dW =
dZ


(2 1)2


Fa2 Z2)(1 a2z2)] 2


The integration which yields W(Z) introduces elliptic functions (see
section 3.1.1.8); on the other hand, it will (now) be possible to cal-
culate U(Z). We note beforehand that, according to the preceding for-
mula, W(Z) assumes the same value on the two edges of the cut and
that, consequently, this solution corresponds to the lifting problem.





N:ACA TM 1354

The relationships of compatibility show that


dU 2X1
dZ B


z(z2 )

(a2 2) (1 a2z2)] 2


U(z) = 2X1
We s + 1)2 ka


We still have to calculate 1 a


Z2 + 1

Z2)(1 a2Z2 2

s a function of w,


(III.13)



For this


purpose, one may write


-wO = d dZ = iil
0 dZ


We put in this integral Z = iu


T (z2 i)-
Z[ 2 1)2d
[a2 2)(1 a2Z2) 2


Wo = 1 f (+ = Ull(a)

S a2 + u2)( + a2u2) 2


The calculation of I(a) can be made with the aid of the function E
(see ref. 24). We shall put


u+ 2
u t

After a few calculations one obtains


I(a) =4 dt
+ (a t2 2 2 1)2t]2
0 1 .


and hence






NACA TM 1354


Finally, the change in variable


sin = t(a2 + 1)
4a2 + (a2 )2t2


shows that if one puts


1 a2
1 + a2


a2(a2 + 1) O


dT = 1
a2(a2 + 1)


Hence the new formula for U(Z)


u(z) 2 a20w
a (a2 + )E1 + a2


(a2


Z2 + 1
1
_Z2)(1 a272) 2


We still have to
One has (fig. 32)


connect a


and Z to the geometrical quantities.


2a
1 + a2


- 3 tan i'0


2X
1 + X2


One puts


t tan u
tan nLQ


and obtains


w0 tan l 0 1

EL 02tan2 o] t2


E a
1 + a
fi-^
li<*^


(III.14)


= 0 tan J